Error estimates for the Scaled Boundary Finite Element Method

Error estimates for the Scaled Boundary Finite Element Method

Karolinne O. Coelhoa,∗, Philippe R. B. Devlooa, Sonia M. Gomesb

a FEC - Universidade Estadual de Campinas, R. Josiah Willard Gibbs 85 - Cidade Universitaria,Campinas, SP, CEP 13083-839, Brazil

b IMECC - Universidade Estadual de Campinas, Campinas, SP, Brazil

Abstract

The Scaled Boundary Finite Element Method (SBFEM) is a technique in which approx-imation spaces are constructed using a semi-analytical approach. They are based onpartitions of the computational domain by polygonal/polyhedral subregions, where theshape functions approximate local Dirichlet problems with piecewise polynomial tracedata. Using this operator adaptation approach, and by imposing a starlike scaling re-quirement on the subregions, the representation of local SBFEM shape functions in radialand surface directions are obtained from eigenvalues and eigenfunctions of an ODE sys-tem, whose coefficients are determined by the element geometry and the trace polynomialspaces. The aim of this paper is to derive a priori error estimates for SBFEM’s solutionsof harmonic test problems. For that, the SBFEM spaces are characterized in the contextof Duffy’s approximations for which a gradient-orthogonality constraint is imposed. Asa consequence, the scaled boundary functions are gradient-orthogonal to any function inDuffy’s spaces vanishing at the mesh skeleton, a mimetic version of a well-known propertyvalid for harmonic functions. This orthogonality property is applied to provide a prioriSBFEM error estimates in terms of known finite element interpolant errors of the exactsolution. Similarities with virtual harmonic approximations are also explored for the un-derstanding of SBFEM convergence properties. Numerical experiments with 2D and 3Dpolytopal meshes confirm optimal SBFEM convergence rates for two test problems withsmooth solutions. Attention is also paid to the approximation of a point singular solutionby using SBFEM close to the singularity and finite element approximations elsewhere,revealing optimal accuracy rates of standard regular contexts.

Keywords: Scaled boundary finite element method, a priori error estimates, Duffy’sapproximations

1. Introduction

The Scaled Boundary Finite Element Method (SBFEM) is a Galerkin method inwhich the approximation spaces are constructed using a semi-analytical approach [1,2, 3, 4]. They are based on general partitions of the computational domain by polygo-nal/polyhedral subregions S (called S-elements), which are supposed to verify the starlikescaling requirement such that any point on the boundary ∂S can be directly visible froma center point (scaling center). The shape functions are computed by the application ofthe scaled boundary technique, involving a specific parametrization of the S-elements,

∗Corresponding author - Phone +55 19 3521-1149Email addresses: [email protected] (Karolinne O. Coelho ), [email protected] (

Philippe R. B. Devloo ), [email protected] ( Sonia M. Gomes )

Preprint submitted to Elsevier December 29, 2020

arX

iv:2

012.

1341

8v1

[m

ath.

NA

] 2

4 D

ec 2

020

which is possible thanks to their scaling property. In classical FE methods, the localapproximations are (mapped) polynomials, and these are known to fail or have very lowconvergence rates when the exact solutions can not be properly represented by polyno-mials. In SBFEM, discretization by piecewise polynomials only takes place at ∂S, whilstthe functions are constructed by approximating local Dirichlet problems internally to S.The method is discussed in the books [3, 4], and articles therein cited.

This incorporation of analytic knowledge about the local behavior of the exact solutionin the approximation spaces is the main property of SBFEM. Therefore, it can be viewedas an operator adapted method. As discussed in [5, 6], in the context of the Partitionof Unity Method, these methods can be expected to perform better when compared withstandard polynomial based FE approximations. They reduce the number of degrees offreedom significantly and hence the computational cost, while improving the quality ofthe solutions. For the case of SBFEM, where only boundary values of the subdomains arediscretized by local surface polynomials, its operator adapted approach revealed itself to beparticularly efficient to approximate problems with stress singularities, such as crack tips,v-notches, and re-entrant corners to name a few applications in elasticity [7, 8, 9, 10, 11].More recently, the method has been applied to highly irregular and heterogeneous domainsdue to the flexibility in generating SBFEM meshes [12, 13]. For instance, the SBFEMhas been applied in quadtree and octree meshes since hanging nodes can be avoided dueto the flexible topology of SBFEM polygonal/polyhedral subregions [14, 15, 10].

The aim of this paper is to derive a priori error estimates of SBFEM approximationsfor the case of Laplace’s equation. Although numerical experiments in the literature pointthat optimal rates of convergence are obtained using SBFEM approximations [16, 17], themathematical demonstrations that give support to the observed numerical results are newcontributions of the current work. For that, we explore two different aspects of SBFEMspaces, shared with Duffy’s approximations [18] or with virtual harmonic spaces [19].

Taking advantage of the scaling property, functions can be represented in the S-elements by coordinates in radial and surface directions. Their values on the boundary∂S live in piecewise polynomial trace spaces, which are radially extended to the interior ofthe subdomain. Therefore, this property puts SBFEM’s spaces in the context of Duffy’sapproximations [18], whose definitions are summarized in Section 2. Partitions of S areobtained by a geometric transformation collapsing a reference quadrilateral, hexahedronor prism on triangular, pyramidal or tetrahedral elements K ⊂ S, each one sharing thescaling center as a vertex (see Section 3.2).

For the model Laplace problem under consideration, the focus of the SBFEM op-erator adapted approach is the approximations inside S-elements by “radial harmonicextensions” of surface components. It is shown that SBFEM’s spaces are Duffy’s approx-imations constructed to solve Laplace problems with piecewise polynomial Dirichlet dataover ∂S. SBFEM spaces are characterized by the enforcement of a gradient-orthogonalityconstraint with respect to Duffy’s approximations vanishing on ∂S and at the center point,as demonstrated in Section 3.3. This perspective on SBFEM approximations reveals thatthe local scaled boundary shape functions are constructed based on an orthogonality con-dition. By enforcing these intrinsic orthogonality constraints, their parametrization inradial and surface directions emerge from the eigenvalues and eigenfunctions of an ODEsystem, whose coefficients are determined by the element geometry and the trace poly-nomial spaces. The scaled boundary functions are gradient-orthogonal to an extendedclass of Duffy’s functions that vanish at the mesh skeleton. It can be viewed as a mimeticversion of a well-known property valid for harmonic functions. These aspects are stated

2

in Proposition 3.1 and used as a key tool to the development of energy error estimates forthe SBFEM in terms of FE interpolation errors in Section 5, as shown in Theorem 5.1,one of the main contributions of this study.

SBFEM also has close similarities with virtual harmonic approximation spaces recentlyintroduced in [19], as explored in Section 4.3 and summarized in Theorem 5.2. In bothcases the trace functions are piecewise polynomials defined over subregion boundaries∂S, which are extended to the interior of S by solving local Dirichlet Laplace problems:whilst the functions in the local virtual spaces are strongly harmonic, in SBFEM spacesthis property is enforced in a reduced extent. Thus, SBFEM approximation errors maycome from the trace polynomial interpolation or by their deviation of being harmonic.However, unlike for the virtual harmonic subspaces, it is possible to explore the radialDuffy’s structure to explicitly compute SBFEM shape functions.

In Section 6, we present results of SBFEM computational simulations for some har-monic test problems confirming the predicted theoretical convergence results of Section 5.We consider 2D and 3D cases with smooth solutions, and discretizations based on differ-ent S-partition geometry, which are formed by internally collapsed triangular, pyramidal,or tetrahedral elements. In the same section, we present p-convergence histories verify-ing asymptotic exponential convergence rates in terms of degrees of freedom (DOF), andcompare results with respect to the ones given by usual FE methods based on partitionsobtained by the conglomeration of the internal collapsed elements. We also pay attentionto the approximation of a singular problem where the singularity occurs by the change ofboundary condition and observe that optimal rates of convergence holds using few DOF,using SBFEM to resolve the singularity. We draw some concluding remarks in Section 7.

2. Duffy’s approximations in triangles, pyramids and tetrahedra

Duffy’s transformations [18] (also referred to as collapsed coordinate systems) areinvertible maps of a rectangle into a triangle, a hexahedron to a pyramid, or a prism to atetrahedron. These maps were originally proposed for integration of vertex singularitiesand they are widely applied to define integration quadrature formulae in triangles [20, 21].Duffy’s transformations are also the basic tools for the construction of spectral methodson simplices (triangles, tetrahedra) [22]. Collapsed isoparametric elements parametrizedby Duffy’s transformations also have applications in crack problems [23, 24, 25].

2.1. Duffy’s geometric transformations

The master elements to be considered have the general form K = [0, 1] × L ⊂ Rd,where L ⊂ Rd−1, d = 2, 3. In the parametric coordinates x = (ξ,η) ∈ K, ξ plays therole of radial variable, and η refers to surface coordinates. The geometry of the masterelements may be one of the following kinds:

• Rectangle K, where L = I is the interval I = [−1, 1].

• Hexahedron K, where L = Q is the rectangle Q = [−1, 1]× [−1, 1].

• Prism K, where L = T is the triangle T = η = (η1, η2); 0 ≤ ηi ≤ 1, η1 + η2 ≤ 1.

The key aspect of geometric Duffy’s transformations FK : K → K is the collapse of onefacet in K on a single vertex of the deformed element K. These maps are also referred in

3

the literature as collapsed coordinate systems [22]. If x denotes the Cartesian coordinatein K, the mapped points x = FK(ξ,η) ∈ K are generically defined by

FK(ξ,η) = ξ (FL(η)− a0) + a0, (1)

where a0 is a vertex in K, and L ⊂ ∂K refers to a facet opposite to a0, which is supposedto be mapped by the geometric transformation FL : L→ L. Notice that the whole facet(0,η),η ∈ L ⊂ K is collapsed over the vertex a0 ∈ K, so that K can be regarded asa quadrilateral with two identical vertices, a hexahedron with four equal vertices, or aprism with three identical vertices. That is why a0 is called the collapsed vertex. Themapping FK can also be seen as a scaling from a point FL(η) ∈ L to the vertex a0. Thisprocess generates radial lines [a0, FL(η)] = a0 + ξr(η), where r(η) = FL(η)− a0.

The Jacobian matrix JK = ∇xFK of the transformation (1) is

JK(ξ,η) =[FL(η)− a0 ξ∇ηFL(η)

]= JK(1,η)

[1 00 ξId−1

], (2)

where Id−1 is the d− 1× d− 1 identity matrix, and JK(1,η) =[FL(η)− a0 ∇ηFL(η)

]is the Jacobian matrix at the surface points where FK(1,η) = FL(η). Thus

J−1K =

(1 00 1

ξId−1

)JK(1,η)−1. (3)

In the following, the geometric transformation (1) is illustrated for the three differentelement geometries considered in the current study.

Case 1: quadrilateral K to triangular K

Let K be the rectangular master element with vertices listed in the next table

a0 a1 a2 a3

(0,−1)(1,−1)(1, 1)(0, 1)

and consider a general triangular element, with vertices a0 = FK(a0), a1 = FK(a1), anda2 = FK(a2), as illustrated in Figure 1. Notice that the edge [a0, a3] collapses ontothe vertex a0 = FK(a0) ∈ K, whilst a1 and a2 are the vertices of the opposite edgeL = FK(1, η) = FL(η).

Figure 1: Geometric illustration of Duffy’s transformation over a triangle as a collapsed quadrilateral.

4

Case 2: hexahedral K to pyramidal K

The master element is the hexahedron K whose vertices are listed bellow.

a0 a1 a2 a3 a4 a5 a6 a7

(0,−1,−1)(1,−1,−1)(1, 1,−1)(1, 1, 1)(1,−1, 1)(0,−1, 1)(0, 1, 1)(0, 1,−1)

Figure 2 illustrates a mapped pyramid with vertices ai = FK(ai), i = 0, · · · 4, a0 beingthe collapsed vertex with opposite quadrilateral face L = [a1, a2, a3, a4]. Observe that:

1. The rectangular face [a0, a5, a6, a7] collapses onto a0;

2. The face [a0, a1, a4, a5] collapses onto the triangle [a0, a1, a4];



5. The face [a3, a6, a5, a4] collapses onto the triangle [a0, a4, a3].

Figure 2: Geometric illustration of a Duffy’s transformation over a pyramid as a collapsed hexahedron.

Case 3: prismatic K to tetrahedral K

The master element is the prism K whose vertices are listed bellow.

a0 a1 a2 a3 a4 a5

(0, 0, 0)(1, 1, 0)(0, 1, 0)(0, 1, 1)(1, 0, 0)(0, 0, 1)

In the tetrahedron shown in Figure 3, with vertices ai = FK(ai), i = 0, · · · 3, the collapsedvertex is a0 and the opposite quadrilateral face is L = [a1, a2, a3]. Note that:

1. The triangular face [a0, a4, a5] collapses onto the vertex a0;

2. The quadrilateral face [a0, a4, a2, a3] collapses onto the triangle [a0, a2, a3];

3. The quadrilateral face [a0, a3, a1, a5] collapses onto the triangle [a0, a1, a3];

4. The quadrilateral face [a1, a2, a4, a5] collapses onto the triangle [a0, a1, a2].

We recall that a hexahedron to tetrahedron Duffy’s transformation can also be derived,as adopted in [22], first via a preliminary step hexahedron to prism, and then the prismto tetrahedron described above.

5

Figure 3: Geometric illustration of a Duffy’s transformation over a tetrahedron as a collapsed prism.

2.2. Duffy’s approximations

Duffy’s approximations refer to functions φ = FK(φ) defined in K and obtained back-tracking functions φ(ξ,η) defined in K, meaning that

φ(x) = φ(ξ,η), for x = FK(ξ,η) ∈ K.

The focus of this paper is on functions φ obtained by separating variables in φ(ξ,η) =ρ(ξ)α(η), where ρ(ξ) is called the radial component, and α(η) is the surface component.It is clear that constant functions in K are mapped to constant functions in K. It shouldalso be noted that for the cases where α(η) is not a constant function, the well definitionof φ at the collapsed point a0 requires that ρ(0) = 0, so that φ(a0) = 0.

We consider function spaces

Dk(K) = φ(ξ,η) = ρ(ξ)α(η); α(η) ∈ Vk(L),

where the surface components α(η) ∈ Vk(L) used to define FE approximation spacesVk(L) = FL(Vk(L)), are finite dimensional polynomial spaces Vk(L). The following casesshall be studied:

1. Vk(L) = Pk(L), polynomials of total degree not greater than k, for the intervalL = [−1, 1] or for the triangle L = T .

2. Vk(L) = Qk,k(L), polynomials of degree not greater than k on each coordinate η1, η2,

for the quadrilateral L = Q.

Gradient operation in Dk(K)

We restrict the study to mapped spaces Dk(K) = FK(Dk(K)) ⊂ H1(K). For instance,as already observed in [26] for the case of triangular elements K, H1(K) corresponds toH1ω(K) where H1

ω(K) := φ ∈ L2ω(K) : ∂η φ ∈ L2

ω−1(K) and ∂ξ φ ∈ L2ω(K), where

ω(ξ,η) = ξ|JK(1,η)|. Particularly, ∂η φ(0,η) = 0 for bounded ∂yφ(x, y).The chain rule implies that

∇xφ(x) = [JK(1,η)]−T[

1 00 1

ξ Id−1

] [ρ′(ξ)α(η)ρ(ξ)∇ηα(η)

]= [JK(1,η)]−T

[ρ′(ξ)α(η)

1ξ ρ(ξ)∇ηα(η)

]= [JK(1,η)]−T

[α(η) 0

0 ∇ηα(η)

] [ρ′(ξ)1ξ ρ(ξ)

]. (4)

If α(η) =∑

l αlN l

k(η) is a linear combination of FE shape functions N lk(η) forming a

basis for Vk(L), then

∇xφ(x) =∑l

αl[B1l(η) B2l(η)

] [ ρ′(ξ)1ξρ(ξ)

], (5)

6

where

B1l(η) = [JK(1,η)]−T[N lk(η)0

], and B2l(η) = [JK(1,η)]−T

[0

∇ηNlk(η)

]. (6)

Special case: α(η) ≡ 1

Let us consider the particular cases of φ(x) ∈ Dk(K), for which φ(ξ,η) = ρ(ξ),meaning that α(η) ≡ 1. A closer look on formula (4) reveals that

∇xφ(x) = [JK(1,η)]−T ρ′(ξ). (7)

For affine elements K and ρ(ξ) = ξ, the mapped function has constant gradient normalto L, so that φ ∈ H1(K) is an affine function vanishing at the collapsed vertex a1, andconstant unitary values φ|L ≡ 1 over the facet L opposite to a1.

3. SBFEM spaces in the context of Duffy’s approximations

Our purpose in this section is to summarize the main aspects of SBFEM approximationspaces under the point of view of Duffy’s approximations and to prove some of theirorthogonality properties to a large range of H1-conforming functions.

3.1. S-elements

The SBFEM adopts macro partitions T = S of the computational domain Ω ⊂ Rd

by subregions S verifying the starlike scaling requirement that any point on the boundaryof S should be directly visible from a point O ∈ S, called the scaling center. We restrictthe study to convex polytopal S-elements (polygonal or polyhedral with flat facets Le).In the literature covering this method, the set ΓS = ∪eLe, e = 1, · · · , NΓS

is known asthe scaled boundary element. A conformal sub-partition T S = Ke of S is formed bysectors Ke sharing the scaling center O as one of their vertices, Le being the facet of Ke

opposite to the scaling center. As illustrated in Figure 4, the sectors Ke may have differentgeometry: triangular in 2D, pyramidal, or tetrahedral in 3D, the facets Le being a linesegment, a quadrilateral or a triangular element, respectively. Moreover, we notice thata three-dimensional S-element may also be partitioned by hybrid tetrahedral-pyramidalmeshes, combining elements of different geometry, with scaled boundary ΓS formed bytriangular-quadrilateral facets. For simplicity, we shall restrict the analysis to partitionsT S where all elements Ke have the same geometry.

This scaled geometry of S implies that the points x ∈ S can be uniquely representedby a radial coordinate 0 ≤ ξ ≤ 1 and a surface coordinate xb. The radial coordinate (orscaling factor) points from the scaling center (ξ = 0) to a point xb ∈ ΓS (where ξ = 1).The geometry of S may also be defined in each sector Ke ∈ T S by a transformation fromthe cartesian coordinates x ∈ Ke to parametric Duffy’s coordinates (ξ,η) ∈ K = [0, 1]×L.This correspondence defines a geometric mapping FKe : K → Ke in the class of Duffy’stransformations described in the previous section, where Ke is interpreted as a collapsedquadrilateral, hexahedral or prismatic geometric element for which the facet FKe(0,η) iscollapsed on top of its vertex x0 in the scaling center O. The points xb in the opposed facetLe are expressed as FKe(1,η) = FLe(η), η ∈ L. For hexahedral or prismatic referenceelements K, the lateral quadrilateral faces are collapsed on triangular faces to form apyramid or a tetrahedron, respectively. These maps are illustrated in Figure 4.

7

.

Figure 4: Illustration of macro partitions T = S, with focus on a sector K ∈ T S , with correspondingDuffy’s transformation, for triangular, pyramidal and tetrahedral K.

3.2. Duffy’s spaces in S-elements

There are two stages in the construction of approximations on polytopal elements S:

1) Definition of a trace space over the boundary ΓS.

2) Extension of the traces to the interior of S.

The first stage is typical of FE contexts, but for specific scaled S-elements the extensionto the interior can be performed in the radial direction.

Trace FE space over the scaled boundary ΓS

Let Λk(ΓS) = C(ΓS) ∩

∏Le⊂ΓS Vk(L

e) be a FE space defined over ΓS. Recall that

Vk(Le) = FLe(Vk(L)), where Vk(L) is the polynomial space considered in L. Let N l,e

k =

FLe(N lk) be shape functions for the local FE spaces Vk(L

e) over the facets Le ⊂ ΓS

obtained backtracking polynomial shape functions N lk for the reference polynomial space

Vk(L). Thus, if α ∈ Λk(ΓS) and xb = FLe(η) ∈ Le, then α(xb) = αe(η) =

∑l α

l,eN lk(η).

As usual, shape functions Nn,Sk (x) for Λk(Γ

S) (say, of cardinality N S) can be obtained by

the assembly of the local shape functions N l,ek , and the functions α ∈ Λk(Γ

S) can globally

represented by linear combinations α(xb) =∑NS

n=1 αnNn,S

k (xb),xb ∈ ΓS. By collecting the

shape functions and multiplying coefficient in N S-vectors NS = [Nn,Sk ] and α = [αn], we

may use the alternative expression α = NS · α.

Radial extensions: Duffy’s space over S

Given a trace function α ∈ Λk(ΓS), take a radial function ρ(ξ), 0 ≤ ξ ≤ 1, to induce

the definition of a function φ(x) by radial extension to the interior of S. Inside each

8

sector Ke ∈ T S and over Le, consider the parametrizations x = FKe(ξ,η) ∈ Ke andxb = FLe(η). Recall the representation α(xb) = αe(η) to define the radial extension

φ(x) = φe(ξ,η) := ρ(ξ)αe(η).

Notice that the surface component αe(η) varies over the partition T S, whilst the radialcomponent ρ(ξ) is the same in all sectors Ke.

Thus, we are in the following context of Duffy’s approximation spaces

Dk(S) =φ ∈ H1(S);∃ φ ∈ Dk(K) such that φ|Ke = FKe(φe),∀Ke ∈ T S

, (8)

where Dk(K) is a given reference Duffy’s approximation space in the master elementK described in Section 2.2. For instance, D0(S) corresponds to the class of functionsin association with φe(ξ,η) = Cρ(ξ),∀Ke, obtained from constant trace functions α ≡C ∈ Λ0(ΓS), where Λ0(ΓS) are the functions with constant value on ΓS. It is clear thatD0(S) ⊂ Dk(S),∀k ≥ 0. Particularly, let us also consider the subspace D0

0(S) ⊂ D0(S)associated to radial functions ρ(ξ) vanishing at ξ = 1.

So far, Dk(S) is a functional space of infinite dimension, for discretization only happensfor the surface component, living in a finite dimensional trace FE space Λk(Γ

S), whilst theradial component can be chosen arbitrarily. The SBFEM spaces to be considered in Sec-tion 3.3 are examples of finite dimensional subspaces of Dk(S). Other finite dimensionalsubspaces Dk,m(S) ⊂ Dk(S) are also of interest: functions φ having local components

φ|Ke = FKe(φe), where φe(ξ,η) = ρ(ξ)αe(η) with ρ ∈ Pm[0, 1] and αe ∈ Vk(L).

Gradient inner product in Dk(S)

Let a pair of functions φ, ψ ∈ Dk(S) with local components φ|Ke = FKe(φe), andψ|Ke = FKe(ψe), φe(ξ,η) = ρ(ξ)αe(η), and ψe(ξ,η) = σ(ξ)µe(η) being associated withradial ρ(ξ), σ(ξ) and surface α(η), µ(η) components. Recalling the trace representationα(xb) = αe(η) =

∑l α

el N

lk(η) for xb ∈ Le, then formula (5) becomes

∇xφ(x) =∑l

αl,e[Be

1l(η) Be2l(η)

] [ ρ′(ξ)1ξρ(ξ)

], for x ∈ Ke,

both d× 1 matrices

Be1l(η) = JKe(1,η)−T

[N lk(η)0

], Be

2l(η) = JKe(1,η)−T[

0

∇ηNlk(η)

]depending on the geometry of the element at the boundary, and on the surface component,but being independent of the radial coordinate ξ (see [4] for the occurrence of thesematrices in the formulation of SBFEM methods). Analogous formula holds for ψ:

∇xψ(x) =∑m

µm,e[Be

1m(η) Be2m(η)

] [ σ′(ξ)1ξσ(ξ)

].

Thus, if 〈φ, ψ〉∇,Ke :=∫Ke∇xφ(x) · ∇xψ(x), dKe, then

〈φ, ψ〉∇,Ke =∑l,m

µm,e αl,e∫ 1

0

∫ 1

−1

[Be

1l Be2l

] [ ρ′(ξ)1ξ ρ(ξ)

]·[Be

1m Be2m

] [ σ′(ξ)1ξ σ(ξ)

]ξd−1|JKe(1,η)| dηdξ

=∑l,m

µm,e αl,e∫ 1

0

∫ 1

−1

[ρ′(ξ) 1

ξ ρ(ξ)]·([BeT

1l

BeT2l

] [Be

1m Be2m

]|JKe(1,η)|

)[σ′(ξ)

1ξ σ(ξ)

]ξd−1 dηdξ

=∑l,m

µm,e αl,e∫ 1

0

[ρ′(ξ) 1

ξ ρ(ξ)]Eeml

[σ′(ξ)

1ξ σ(ξ)

]ξd−1dξ, (9)

9

where the entries in the matrix Ee

ml=

[Ee

11,ml Ee12,ml

Ee21,ml Ee

22,ml

]are

Ee11,ml =

∫ 1

−1

BeT1l (η)Be

1m(η)|JKe(1,η)| dη, Ee12,lm =

∫ 1

−1

BeT1l (η)Be

2,m(η)|JKe(1,η)| dη.

Ee21,ml =

∫ 1

−1

BeT2l (η)Be

1m(η)|JKe(1,η)| dη, Ee22,lm =

∫ 1

−1

BeT2l (η)Be

2,m(η)|JKe(1,η)| dη.

3.3. SBFEM spaces in S-elements

There are two stages in the construction of local SBFEM approximation spaces inS-elements, that we shall denote by Sk(S): the restriction of a function in Sk(S) overthe scaled boundary ΓS is set in the FE trace space Λk(Γ

S), and in the radial direction,it is obtained analytically in terms of eigenvectors and eigenfunctions of an ODE sys-tem, known SBFEM equation. Our purpose is to highlight the main aspects of SBFEMspaces in the context of Duffy’s approximations Dk(S) for S-elements, and to show thata paramount for the derivation of the SBFEM equation is the enforcement of a gradientorthogonality constraint.

Precisely, having in mind that our goal is the solution of harmonic model problems,let us define the subspace

Sk(S) =

φ ∈ Dk(S); 〈φ, ψ〉∇,S :=

∫S∇xφ(x) · ∇xψ(x) dS = 0, ∀ψ ∈ D0

0(S), ψ(O) = 0

. (10)

This definition suggests that the functions φ ∈ Sk(S) ⊂ Dk(S) have boundary valuesφ|ΓS = α ∈ Λk(Γ

S), and they are “weak solutions” of the harmonic equation ∆Φ = 0in S with Dirichlet data α. Thus, in some extent, Sk(S) can be interpreted as “radialharmonic extensions” of the trace FE space Λk(Γ

S) to the interior of S.Notice that φ0(x) ≡ 1 is clearly in Sk(S). The goal is to construct linearly independent

shape functions φi ∈ Sk(S) such that

Sk(S) = span φi.

It is known that the radial components and boundary values for the SBFEM shape func-tions φi are determined by a particular family of exact eigenvalues and eigenfunctionssolving an ODE system [4]. Next, we recover this representation of φi using the currentapproach of Duffy’s approximations constrained by the gradient orthogonality propertyexpressed in (10).

Recall that, as a function in Dk(S), the shape function φi ∈ Sk(S) must be ob-tained as φi|Ke = FKe(φei ), backtracking a function φei (ξ,η) = ρi(ξ)α

ei (η) ∈ Dk(K).

Moreover, we are assuming that the local surface components αei (η) have expressionsαei (η) =

∑l α

l,ei N

lk(η), as linear combinations of shape functions N l

k(η) ∈ Vk(L). Thus, it

is necessary to characterize the radial functions ρi(ξ) and the multiplying coefficients αl,eiallowing the verification of the gradient orthogonality property stated in definition (10).

Derivation of the SBFEM equation

Let ψ(x) ∈ Dk(S) be a general function locally defined as ψ|Ke = FKe(ψe), whereψe(ξ,η) = σ(ξ)µe(η) ∈ Dk(K) and consider its gradient inner product

〈φi, ψ〉∇,S =

∫S

∇xφi(x) · ∇xψ(x) dS =∑e

〈φi, ψ〉∇,Ke ,

10

with a (searched) shape function φi ∈ Sk(S), where the terms 〈φi, ψ〉∇,Ke are expressedas in (9). In fact, this formula can be rewritten as:

〈φi, ψ〉∇,Ke =∑m,l

µm,e αl,e∫ 1

0

(ξd−1ρ

′

i(ξ)Ee11,mlσ

′(ξ) + ξd−2ρ

′

i(ξ)Ee12,mlσ(ξ)

+ξd−2ρi(ξ)Ee21,mlσ

′(ξ) + ξd−3ρi(ξ)E

e22,mlσ(ξ)

)dξ. (11)

Let us denote by Ers

, r, s ∈ 1, 2, the N S × N S matrices obtained by assembling thematrices Ee

rs,ml, element-by-element, according to the interelement connectivity. The

process is similar to matrix assembly for FE discretizations of boundary problems in Rd−1.Moreover, consider the vector functions Φi(ξ) = ρi(ξ)αi, and Ψ(ξ) = σ(ξ)µ collectingboth radial and trace information of the shape functions φi(x) and of test functions ψ(x).Applying this notation, and summing up the contributions in (11), we obtain

〈φi, ψ〉∇,S =

∫ 1

0

Ψ′

(ξ) ·[ξd−1E

11Φ′

i(ξ) + ξd−2E21

Φi(ξ)]

+

Ψ(ξ) ·[ξd−2E

12Φ′

i(ξ) + ξd−3E22

Φi(ξ)]dξ. (12)

Consider Qi(ξ) =

[ξd−1E

11Φ′

i(ξ) + ξd−2E21

Φi(ξ)], and apply integration by parts to ob-

tain ∫ 1

0

Ψ′

(ξ) · Qi(ξ)dξ = Ψ · Q

i

]1

0−∫ 1

0

Ψ(ξ) · Q′

i(ξ) dξ. (13)

For Q′

i(ξ) = E

11

(ξd−1Φ

′′

i (ξ) + (d− 1)ξd−2Φ′

i(ξ))

+ E21

(ξd−2Φ

′

i(ξ) + (d− 2)ξd−3Φi(ξ))

,

the inclusion of formula (13) in (12) gives

〈φi, ψ〉∇,S = Ψ · Qi

∣∣∣10−∫ 1

0

Ψ(ξ)·[ξd−1E

11Φ′′

i (ξ) +[(d− 1)E

11− E

12+ E

21

]ξd−2Φ

′

i(ξ)

+[(d− 2)E

21− E

22

]ξd−3Φi(ξ)

]dξ. (14)

Recall that the purpose is to characterize the functions φi(x) ∈ Dk(S) such that theorthogonality property Mi = 0 holds for all functions ψ ∈ D0

0(S), i.e., vanishing on ΓS,but also vanishing on the scaling center. That is, for σ(0) = σ(1) = 0 and consequentlyΨ(0) = Ψ(1) = 0. These constraints on ψ cancel the boundary term in (14). On the otherhand, the condition for vanishing the integral term in (14) for all Ψ(ξ) is equivalent tosay that Φ(ξ) must solve the following equation

ξd−1E11

Φ′′

i (ξ) +[(d− 1)E

11− E

12+ E

21

]ξd−2Φ

′

i(ξ) +[(d− 2)E

21− E

22

]ξd−3Φi(ξ) = 0.

(15)Notice that this is the usual scaled boundary equation documented in [4] for the SBFEMshape functions. The resolution of (15) is well documented in the SBFEM literature, andit involves an auxiliary eigenvalue problem for an ODE system in terms of both Φi(ξ) andQi(ξ). For self completeness, the methodology is briefly described in Appendix A.

In summary, the resulting solutions Φi = ρi(ξ)αi have the form ρi(ξ) = ξλi , andαi = Ai, where λi and Ai refer to positive real parts of the eigenvalues and the associ-ated eigenfunctions for the ODE system equivalent to the SBFEM equation (15). This

11

information is required for the construction of the SBFEM basis functions φi, giving theradial components ρi(ξ) and the trace surface components αi ∈ Λk(Γ

S) recovered fromthe coefficient vectors αi. Thus, the corresponding expressions are

φi(x) = φei (ξ,η) = ξλi∑l

Al,ei Nlk(η), for x = FKe(ξ,η) ∈ Ke. (16)

Analogously, associated to Qi(ξ) are the flux functions

qi(x) = qei (ξ,η) = ξλi∑l

Ql,e

iN lk(η), for x = FKe(ξ,η) ∈ Ke. (17)

3.4. Orthogonality properties of the SBFEM spaces

In this section, we highlight two kinds of gradient orthogonality properties held by theSBFEM approximation spaces.

Intrinsic gradient orthogonality property for Sk(S)

The usual procedure for the construction of SBFEM shape functions is the determi-nation of analytic eigenfunctions for the SBFEM equation (15). We have shown in theprevious section that there is another characterization of these shape functions that arenot well recognized. Namely, implicit in the condition for a function φ ∈ Dk(S) to solvethe SBFEM equation (15) is the gradient orthogonality property, enforced from the start,in the definition of the subspaces Sk(S) in (10). Precisely, a function φ ∈ Sk(S) ⊂ Dk(S)if the gradient orthogonality constraint

〈φ, ψ〉∇,S =

∫S

∇xφ(x) · ∇xψ(x) dS = 0 (18)

holds for all ψ ∈ D00(S), with ψ(O) = 0. In such case, then φ solves equation (15).

Extended gradient orthogonality property for Sk(S)

Let H(S) denotes the space of harmonic functions in S. Then, it is clear that〈φ, ψ〉∇,S = 0 for all φ ∈ H(S) and ψ ∈ H1

0 (S), giving the well-known decomposition

H1(S) = H(S)∇⊕H1

0 (S), (19)

where the symbol∇⊕ denotes the orthogonality relation with respect to the gradient inner

product 〈·, ·〉∇,S. Our purpose is to show a similar relation for Duffy’s spaces Dk(S) ⊂H1(S), Sk(S) playing the role of the harmonic functions. For that, we need to extend thegradient orthogonality property (18) to functions ψ ∈ D0(S).

Proposition 3.1. The orthogonality property

〈φ, ψ〉∇,S = 0, ∀φ ∈ Sk(S) and ψ ∈ D0(S) (20)

is valid. Thus,

Dk(S) = Sk(S)∇⊕D0(S) (21)

holds as a mimetic version of (19).

12

Proof. A crucial step in the derivation of the SBFEM equation (15) is the formula forthe gradient inner product 〈φ, ψ〉∇,S given in (14), where two terms enter into play: aboundary term and an integral term. The constraints ψ ∈ D0

0(S) and ψ(O) = 0 make theboundary term to be zero, and (15) derives from the assumption (18).

Now let us relax the constraints ψ ∈ D00(S) and ψ(O) = 0. Instead, take ψ in a

broader space D0(S). Clearly, the property 〈φ, ψ〉∇,S = 0 holds for φ = φ0 ≡ 1. Thus, itis sufficient to verify it for all shape functions φ = φi associated with eigenvalues λi 6= 0.

Notice that the desired orthogonality property (20) is valid for ψ ∈ D0(S), withψe(ξ,η) = Cσ(ξ) in the sectors Ke, if and only if it holds for functions ϕ = ψ −Cσ(1) ∈D0

0(S), i.e., for the cases where ϕ(ξ) = C(σ(ξ)− σ(1)), with ϕ(1) = 0. For them, we applyequation (15), valid for all shape functions φi ∈ Sk(S), to reduce the equation (14) to

〈φi, ϕ〉∇,S = ϕ(1)∑n

Qin

(1)− ϕ(0)∑n

Qin

(0).

Thus, since ϕ(1) = 0 and Qin

(0) = 0, we obtain the orthogonality property (20).

4. Interpolants

When a Galerkin method is used to approximate a boundary value problem, one ofthe most important choices is the family of approximation spaces. For elliptic problemsthe achievable error of approximation is equal to the error obtained by approximating thesolution of the partial differential equation directly from the trial space. The accuracy isaccessed a priori by bounds computed in terms of interpolant errors using the approxima-tion space. In the context of piecewise defined approximations over subregions (elements)of the computational domain, as is the case of FE methods, the interpolants usually showthe following characteristics:

• Locality: in each subregion, a polynomial trace interpolant over the boundary isextended to the interior (a process also called lifting).

• Global conformity: it follows directly from the hypothesis that the trace interpolantsdepend exclusively on the function restriction over subregion boundaries.

• Optimality: optimal interpolation error estimates are achieved with respect to thediscretization parameters: mesh width and polynomial order.

In this direction, the plan is to construct interpolants in SBFEM trial spaces, and toexplore them to evaluate the potential of SBFEM approximations. Firstly, let us introducesome new notation and auxiliary results already known in other contexts.

Consider a family of conformal polytopal partitions T h = S of Ω by S-elements,as described in Section 3.1. Define the mesh skeleton Γh = ∪L∈EhL by the assemblyof all facets (edges of faces) in Eh = L ⊂ Γh,S, S ∈ T h. The parameter h refers tothe characteristic size of the facets in Γh. Moreover, define the conglomerate partitionsPh = ∪S∈T hT h,S of Ω. Recall that the elements K ∈ T h,S may be affine triangles,pyramids, or tetrahedra inheriting the conformal property from T h. In principle, shaperegularity of Ph is not a granted property.

Based on the partitions Γh, T h or Ph, we consider the following approximation spaces.

• FE trace spaces: Λk(Γh) = C0(Γh) ∩

∏L∈Eh Vk(L), piecewise polynomial spaces,

where Vk(L) = Pk(L), for 1D edges and triangular facets L, and Vk(L) = Qk,k(L),for quadrilateral facets L.

13

• Duffy’s spaces Dhk ⊂ H1(Ω): given the local Duffy’s spaces Dhk(S), S ∈ T h definedin Section 3.2, set

Dhk = w ∈ H1(Ω);w|S ∈ Dhk(S), S ∈ T h,D0,h

0 = w ∈ H1(Ω);w|S ∈ D00(S), S ∈ T h.

Notice that D0,h0 ⊂ Dhk ,∀k ≥ 0.

• SBFEM spaces Shk ⊂ H1(Ω): given local SBFEM spaces Shk(S) ⊂ Dhk(S), S ∈ T h,described in Section 3.3, define

Shk = w ∈ H1(Ω);w|S ∈ Shk(S), S ∈ T h,

and set Shk,0 = Shk ∩H10 (Ω).

• FE spaces Vh,FEk ⊂ H1(Ω): Consider the following FE spaces based on the conglom-erated meshes Ph.

1. Triangular (2D) and tetrahedral (3D) meshes Ph: Vh,FEk := Pk(Ph) ∩ H1(Ω),where Pk(Ph) stands for functions piecewise defined by polynomials in Pk(K),K ∈ Ph, of degree not greater than k.

2. Pyramidal (3D) meshes Ph: let us consider Vh,FEk := U (0),k(T h) ∩ H1(Ω),piecewise defined by a class of rational polynomials U (0),k(K), for K ∈ Ph [27].Traces of functions in U (0),k(K) are in Pk(L) for triangular faces, and in Qk,k(L)if L is quadrilateral. Moreover, Pk(K) ⊂ U (0),k(K).

Proposition 4.1. (i) For w ∈ Vh,FEk , w|Γh ∈ Λk(Γh). (ii) Pk(Ph) ⊂ Vh,FEk ⊂ Dhk .

Proof. The trace property (i) and the polynomial inclusion in (ii) are already known. Toproof the second embedding property in (ii), let us start by considering three particularcollapsed triangular, pyramidal and tetrahedral reference elements.

• A triangular reference element K: Let K be the reference triangle, with collapsedvertex a0 = (0, 0), and the opposed edge L = [a1, a2], where a1 = (1, 0) and a2 =(1, 1). Taking the mapping FL : L → L, defined as FL(η) =

(1+η

2, 1−η

2

), the

Duffy’s transformation from K over K becomes x = ξ2(1 + η), y = ξ

2(1− η), whose

inversion is ξ = x + y, η = x−yx+y

. Let ψ ∈ Dk(K) be the pullback of functions

FK(ψ) ∈ Dk(K), where ψ(ξ,η) = ξkα(η), so that ψ(x, y) = (x + y)kα(x−yx+y

). Thus,

by varying α ∈ Pk(L), we conclude that all functions ψ(x, y) ∈ Pk(K) can berecovered in Dk(K).

• A pyramidal reference element: Suppose K is a pyramid with vertex a0 = (0, 0, 1),and opposed face L = [a1, a2, a3, a4], with vertices a1 = (0, 0, 0), a2 = (1, 0, 0),a3 = (1, 1, 0), and a4 = (0, 1, 0). The FE space U (0),k(K) ⊂ H1(K) proposed in [27]is the first space of an exact sequence U (s),k(K) verifying the De Rham commutingproperty. Their definition considers the the geometric transformation S∞ : K∞ → Kof the ”infinite pyramid” K∞ = (x, y, z) ∈ R3;x, y, z ≥ 0, x ≤ 1, y ≤ 1 ∪ ∞,given by S∞(x, y, z) =

(x

1+z, y

1+z, z

1+z

), S∞(∞) = a0. The functions w ∈ U (0),k(K)

are obtained by the pullback S∞(u) of functions u in a properly chosen subspaceof the rational functions Qk,k,k

k (K∞) = q1+z

; q ∈ Qk,k,k(K∞). Our goal is to show

14

that U (0),k(K) can also be interpreted in the context of the Duffy’s space Dk(K).For that, consider the hexahedron H = [0, 1] × [0, 1] × [0, 1], with the coordinatesystem (µ1, µ2, ξ), with (µ1, µ2) ∈ [0, 1] × [0, 1] and 0 ≤ ξ ≤ 1. Observe that thegeometric transformation F∞ : H → K∞, F∞(µ1, µ2, ξ) = (µ1, µ2,

ξ1−ξ ) collapses the

face ξ = 1 in H onto ∞. Moreover, Qk,k,kk (K∞) = F∞(Qk,k,k(H)). Consequently,

U (0),k(K) ⊂ S∞(Qk,k,kk (K∞)) = S∞(F∞(Qk,k,k(H)). (22)

On the other hand, the transformation FK : H → K, defined by the composition x =FK(η1, η2, ξ) = S∞(F∞(η1, η2, ξ)) results to be a Duffy’s transformation collapsingthe face ξ = 1 in H on top of the vertex a0 ∈ K. Consequently, FK(Qk,k,k(H)) ⊂Dk(K). Thus, using (22), we obtain U (0),k(K) ⊂ Dk(K).

• A tetrahedral reference element: Suppose K is the reference tetrahedron with col-lapsed vertex a0 = (0, 0, 0), and opposed quadrilateral face L = [a1, a2, a3], witha1 = (1, 0, 1), a2 = (1, 0, 0) and a3 = (1, 1, 0). Notice that L can be mapped byx = FL(η), where x = 1−η1−η2, y = η1, and z = η2. Then, the Duffy’s transforma-tion is FK(ξ,η) = ξFL(η), whose inverse is ξ = x+y+z, η1 = y

x+y+z, η2 = z

x+y+z

Let ψ = FK(ψ) ∈ Dk(K), with ψ(ξ,η) = ξkα(η), and α ∈ Pk(L). Thus, the func-tions ψ(x, y, z) = (x+ y + z)kα( y

x+y+z, zx+y+z

) recover all functions in Pk(K).

Now consider a general element Ke ∈ Ph, with collapsed vertex O, and opposed faceLe with vertices ael . Notice that Ke can be seen as a geometric affine transformationof one of the reference elements K described above, i.e., Ke = T e(K), such that O =T e(a0), ael = T e(al), and thus Le = T e(L)). Since the polynomials Pk(K), for trianglesand tetrahedra, and rational polynomials Sk(K)), for pyramids, are preserved by affinetransformations, then we conclude that Vh,FEk ⊂ Dhk .

4.1. FE interpolants

Interpolant operators Fh,FEk : Hs(Ω) → Vh,FEk have being designed as useful toolsfor functions in general Sobolev spaces Hs(Ω), s ≥ 1. As already mentioned, they areconstructed by first defining a piecewise polynomial trace interpolant over the facetsL ⊂ ∂K of each element K ∈ Ph, and then by extending this trace interpolant tothe interior of K. Let us recall some examples and error estimates already availablein the literature. For them, we assume the affine conglomerate triangular, pyramidalor tetrahedral partitions Ph are regular (e.g. quasi-uniform and shape regular, withparameters independent of h). Under these circumstances, the following estimates hold.

• There are interpolands Fh,FEk w over FE spaces Vh,FEk = Pk(Ph)∩H1(Ω) defined in[28] for triangles and in [29] for thetrahedra. Suppose w ∈ Hs(Ω), s > 3

2in 2D, and

s > 2 in 3D, then the estimate

|w −Fh,FEk w|H1(Ω) .hµ−1

kd−2‖w‖Hs(Ω)

holds for µ = min(k+ 1, s), where the leading constant on the right side is indepen-dent of w, h, and k (but depends on s and regularity parameters of Ph).

• There are also the projection-based interpolants, proposed by L. Demkowicz andcoworkers, as expounded in [30, 31]. They admit a general form, without requiring

15

any specific geometric aspect, and have the flexibility to treat general local spaces,not necessarily polynomials. Note that such constructions may require additionalregularity assumptions beyond the minimal H1-conformity. Indeed, the trace in-terpolants may require interpolation at element vertices, requiring the regularityH1+s with s > 1/2 in 3D FE settings. For FE spaces Vh,FEk = Pk(Ph) ∩ H1(Ω)based on tetrahedra, the error estimates stated in [31, Theorem 2.2] for projectionbased-interpolants Fh,FEk w have the non-optimal form

|w −Fh,FEk w|H1(Ω) . (ln k)2

(h

k

)d−2

|w|Hs , s > 3/2. (23)

The suboptimal logarithmic factor appearing in (23) can be dropped in the k-versionunder the more stringent regularity assumption s ≥ 2 [32, Corollary 2.12].

• For pyramidal partitions Ph, projection-based interpolants Fh,FEk w over the FEspaces U (0),k(T h) are defined in [27]. However, to the best of our knowledge, errorestimates are still missing for them, but optimal h-convergence rates have beenobserved in numerical experiments presented in [33].

4.2. SBFEM interpolant

As for the cases of FE spaces, we construct interpolant operators Πhk : Hs(Ω) → Shk,

for sufficiently smooth functions w ∈ Hs(Ω), following three steps: a trace interpolantIhk : Hs(Γh)→ Λk(Γ

h), local projections Πh,Sk : Hs(S)→ Sk(S) extending trace functions

to the interior of the element, and assembly of local interpolants.

1. Trace interpolant Ihk : Hs(Ω)→ Λk(Γh) - it is piecewise defined on the facets L ∈ Eh,

following any of the interpolation strategies used so far for the FE spaces Vh,FEk .

2. Local projections Πh,Sk : Hk+1(S)→ Sk(S): Πh,S

k w ∈ Dh,Sk solves the problem

〈Πh,Sk w, v〉∇,S = 0 ∀v ∈ D0

0(S), (24)

Πh,Sk w|ΓS = Ihk w|ΓS . (25)

Notice that equation (24) ensures that Πh,Sk w ∈ Sk(S) and the relation (25) enforces

the trace constraint matching Πh,Sk w to the trace interpolant of w. It is clear from

these equations the interpretation of Πh,Sk as ”radial harmonic extension” of the trace

interpolant Ihk w to the interior of S. Let ωS be the coefficients in the expansion

Ihk w(xb) =∑NS

n=1 ωn,SNn,S

k (xb), xb ∈ ΓS. We seek for coefficients c = [ci] such that

Πh,Sk w =

∑i ciφ

Si ∈ Shk(S). According to the definition of the local spaces Shk(S),

the solution is c = ωSA−1, where A = AS is the eigenvector matrix associated to

the traces of the SBFEM shape-functions φSi over ΓS.

3. Assembly - Define Πhk w by assembling the local contributions Πh

k w|S = Πh,Sk w.

It is clear that Πh,Sk w|L = Πh,S

k w|L over an interface L = S ∩ S ′ shared by twoS-elements. Thus, the conformity property Πh

k w ∈ H1(Ω) holds.

16

Remarks

(1) In the same manner as FE interpolants Fh,FEk w, the SBFEM interpolant Πhk satisfies

the two fundamental properties: locality and global conformity. However, they differon the way the trace interpolant is extended to the interior of the S-elements bytheir local projections. Recall that the ”radial harmonic extension” adopted in theSBFEM context is possible due to the particular scaled geometry of the S-elements.Moreover, when the SBFEM interpolant shares the trace interpolant of Fh,FEk w,then it is clear that

Πhk w = Πh

k Fh,FEk w. (26)

(2) Since Ihk w = w|Γh for functions w ∈ Dhk , the trace constraint (25) means that w −Πhk w ∈ D

0,h0 for all functions w in the Duffy’s space Dhk . Consequently, Proposition

3.1 implies the orthogonality property

〈w − Πhk w, v〉∇ =

∑S∈T h

〈w − Πh,Sk w, v〉∇,S = 0, ∀w ∈ Dhk , ∀v ∈ Shk. (27)

4.3. Comments on the SBFEM interpolation errors

Unlike general purpose FE techniques, SBFEM approximations are constructed tobe applied for a specific type of problem. Thus, for the model Laplace problem underconsideration, there is no interest in accessing the accuracy of SBFEM interpolants Πh

k wwhen applied to other than for harmonic functions w ∈ H(Ω). For them, the sources ofSBFEM interpolation errors are two-fold:

(i) the polynomial discretization of traces w|Γh ≈ Ihk w ∈ Λhk.

(ii) the deviation of Πh,Sk w ∈ Shk of being an harmonic function.

In this direction, let us consider the subspaces

Vh,∆k = w ∈ H(Ω);w|Γh ∈ Λk(Γh),

where only trace discretization takes place. Denoted by harmonic virtual spaces, they havebeen used in the context of the operator adapted virtual FE method proposed in [19], anddesigned to solve two-dimensional harmonic problems. The term “virtual” emphasizesthat functions in Vh,∆k are not known explicitly in the interior of each subregion S ∈ T h.

The finite-dimensional spaces Vh,∆k have close similarities with the SBFEM spaces Shk.In both cases, the trace functions are in Λk(Γ

h), which are extended to the interior of theS-elements by solving local Dirichlet Laplace problems: whilst the functions in the localspaces V ∆

k (S) = Vh,∆k |S are strongly harmonic in S, the ones in Sk(S) are harmonic in aweaker sense. However, unlike for the harmonic subspaces V ∆

k (S), it is possible to explorethe radial Duffy’s structure of Shk(S) to explicitly compute shape functions for them, asdescribed in the previous section.

Let us consider the harmonic virtual interpolant Fh,∆k : Hs(Ω)→ Vh,∆k by solving thelocal Laplace problems

〈Fh,∆k w, v〉∇,S = 0 ∀v ∈ H10 (S), (28)

Fh,∆k w|ΓS = Ihk w|ΓS , (29)

17

where the trace interpolant Ihk w is the one adopted in Πhk w. Note that this is an analytic

recovery problem for it is not directly accessible for computation, whilst the SBFEMinterpolant Πh

k w is a computable recovery problem.For an harmonic function u ∈ H(Ω), let us consider the decomposition

u− Πhk u = (u−Fh,∆k u) + (Fh,∆k u− Πh

k u) = (i) + (ii). (30)

The first term (i) = u−Fh,∆k u compares two harmonic functions differing on the skeletonΓh by the trace interpolation error u−Ihk u, meaning that only the interface errors requireto be estimated. In fact, the application of Neumann trace inequality ([34, TheoremA.33]) in each S-element S ∈ T h gives

|u−Fh,∆k u|H1(S) . ‖u− Ihku‖H 12 (∂S)

. (31)

We refer to [19, Lemma 4.4, Lemma 4.5] for estimates of (31) in the particular Gauss-Lobatto trace interpolation case, and under some specific graded polygonal mesh circum-stances. On the other hand, since

Πhk u = Πh

k Fh,FEk u, (32)

the second term becomes (ii) = Fh,∆k u − Πhk u = Fh,∆k u − Πh

k Fh,∆k u, representing the

SBFEM interpolation error for the harmonic virtual function Fh,∆k u ∈ Vh,∆k . Conse-quently, according to (24) and (28), we obtain

〈Fh,∆k u− Πhk F

h,∆k u, v〉∇ =

∑S∈T h

〈Fh,∆k u− Πhk F

h,∆k u, v〉∇,S = 0, ∀v ∈ D0,h

0 . (33)

In other words, the second term (ii) = Fh,∆k u−Πhk u, which vanishes in Γh, is orthogonal

to D0,h0 with respect to the gradient inner product. Thus its energy norm is a measure of

the deviation of Πhk F

h,∆k u of being an harmonic function. Since Pm(T S)∩H1

0 (S) ⊂ D00(S),

for polynomials of arbitrary degree m ≥ 1, the energy norm of the second term (ii) isexpected to decay exponentially, and it is eventually dominated by the energy norm ofthe trace interpolation error represented by the first term (i).

4.4. Examples of SBFEM interpolation errors in a single S-elementLet us consider some examples to illustrate the accuracy capabilities of the SBFEM

interpolant, both for smooth or boundary point singularity harmonic functions defined ina single S element, using refined scaled boundary elements Γh,S, where uniform Lagrangetrace interpolation is adopted.

Example 1 - SBFEM interpolation of a smooth harmonic function in 2D

In the region S = [−1, 1]× [−1, 1] consider the harmonic function

u(x, y) = exp (πx) sin (πy),

and interpret S as polygonal regions of 4n facets, n = 2, 4 and 8, as illustrated in Figure5. The scaled boundary elements Γh,S are obtained by subdividing each side of ∂S inton subintervals of width h = 2

n. In other words, S is formed by 4n triangles Ke sharing

the scaling center point as a vertex and having one edge in Γh,S as an opposite facet. Thetriangles Ke are mapped by Duffy’s geometric transformations described in Section 2.

For these kinds of scaled geometry, we consider the SBFEM space Sh,Sk , for 1 ≤ k ≤ 4,

and compute the interpolants Πh,Sk u. The corresponding error histories versus h are

plotted in Figure 6, reflecting the usual convergence behavior governed by the FE tracediscretizations Ihk u over ∂S, of order k in the energy norm, and order k+1 in the L2-norm.

18

u(x, y) h = 1 h = 12 h = 1

4

Figure 5: Example 1 - Harmonic function u(x, y) and scaled triangular partitions T h,S of S = [−1, 1]×[−1, 1] with scaled boundary Γh,S formed by 4n uniform facets of width h = 2

n , n = 2, 4 and 8.

Figure 6: Example 1 - Energy and L2 SBFEM interpolation errors versus h: Shk(S) based on the scaled

triangular partitions T h,S of Figure 5, and trace spaces Λh,Sk of degree k = 1, · · · , 6.

Example 2 - SBFEM interpolation of a smooth harmonic function in 3D

The second example is for the harmonic function

u(x, y, z) = 4(

exp(πx

4

)sin(πy

4

)+ exp

(πy4

)sin(πz

4

))defined in the region S = [0, 1] × [0, 1] × [0, 1]. Let SBFEM spaces Sh,Sk obtained byconsidering S as polyhedral regions with 6n2 facets, as illustrated in Figure 14. The scaledboundaries Γh,S are formed by subdividing each face in ∂S into n× n quadrilaterals, andwe set the characteristic size h = 1

n. Thus, the partitions T h,S are composed of 6n2

pyramids Ke sharing the scaling center point as a vertex, which are mapped by Duffy’sgeometric transformations of the reference hexahedron, as described in Section 2. We

h = 1 h = 12 h = 1

4

Figure 7: Example 2 - Scaled pyramidal partitions T h,S of S = [0, 1]× [0, 1]× [0, 1] with scaled boundaryΓh,S formed by 6n2 uniform quadrilateral facets of characteristic width h = 1

n , n = 1, 2 and 4.

approximate u by the SBFEM interpolants Πh,Sk u, and the interpolation error curves are

19

plotted in Figure 8, revealing the typical optimal convergence rates of order k in energynorm, and order k + 1 in the L2 norm of the trace interpolant.


pyramidal partitions T h,S of Figure 7, and trace spaces Λh,Sk of degree k = 1, · · · , 4.

Example 3 - SBFEM interpolation of a singular harmonic function

In the region S = [−1, 1]× [0, 1] define the harmonic function

u(x, y) = 2−1/4

√x+

√x2 + y2 = 21/4

√r cos(

θ

2),

shown in Figure 9, with a radial square root singularity at the boundary point O = (0, 0)(r = 0), caused by boundary condition change from Dirichlet u(x, 0) = 0, for x < 0, to

Neumann ∂u/∂y(x, 0) = 0, for x > 0. This function belongs to H32−ε(Ω), for all ε > 0.

We put the scaling center at the origin and take an open scaled boundary Γh,S overthe two vertical and the top horizontal sides of S, which are uniformly subdivided: nuniform intervals for the vertical edges, and 2n for the top edge, n = 1, 2 and 4. Thisway, in each refinement level, S is composed of internal triangular partition T h,S formedby 4n triangles sharing the scaling center as collapsed vertex, and opposite facet widthh = 1

n. Because Γh,S is not a closed curve, some care must be taken in the construction

of the SBFEM space Shk(S) in order to incorporate boundary data for u on the bottomboundary side of S. This is accomplished by enforcing in the second order SBFEM ODEsystem a vanishing Dirichlet boundary condition on one side (associated with vanishingtrace value at xb = (−1, 0)), whilst a vanishing Neumann condition is assumed on theopposite side (associated with vanishing normal trace at xb = (1, 0)). These boundarydata are radially extended over the sectors [−1, 0) and [0, 1].

SBFEM interpolation errors for this singular example are plotted in Figure 10, reveal-ing usual optimal convergence rates known for trace interpolations by piecewise polyno-mials. These results reflect the role of the two terms in the decomposition (30), where thedominant contribution is expected to come from the virtual interpolant error, determinedexclusively by the trace interpolant, which is not affected by eventual function singularitynot interacting with the scaled boundary Γh,S.

5. Galerkin SBFEM approximations

This section is dedicated to the Galerkin SBFEM for the Laplace’s model problem

∆u = 0, in Ω, (34)

γ0(u) = uD, on ∂Ω,

20

u(x, y)

h = 1 h = 12 h = 1

4 h = 18

Figure 9: Example 3 - Singular harmonic function u(x, y) and scaled triangular partitions T h,S of S =[−1, 1]× [0, 1], with open scaled boundary Γh,S , with 4n uniform facets, h = 1

n , n = 1, 2, 4 and 8.


triangular partitions T h,S of Figure 9, and trace spaces Λh,Sk of degree k = 1, · · · , 4.

where uD ∈ H1/2(∂Ω), and γ : H1(Ω)→ H1/2(∂Ω) is the usual trace operator. We assumethat uD is sufficiently smooth for the definition of the trace interpolant.

Let Shk be the trial SBFEM approximation spaces based on geometric partitionsT h = S of Ω by S-elements, Πh

k : Hs(Ω) → Shk being the corresponding interpolantoperators, as defined in the previous section. The Galerkin SBFEM for problem (34)searches approximate solutions uh ∈ Shk satisfying:

a(uh, v) = 0 ∀v ∈ Shk,0, (35)

uh|∂Ω = Ihk uD|∂Ω, (36)

where a(w, v) :=∫

Ω∇u · ∇v dΩ is the usual bounded symmetric bilinear form for u,w ∈

H1(Ω). The bilinear form a is well known to be coercive, meaning there exist ν > 0such that a(v, v) ≥ ν‖v‖2

H1 , ∀v ∈ H10 (Ω). Thus, problem (35)-(36) is well-posed (see [35,

Proposition 3.26]).

5.1. Error analysis for the SBFEM

For the error analysis of the Galerkin SBFEM discretization (35)-(36), the purpose is toexplore the properties (27) and (33) to estimate energy errors |u−uh|H1 in approximatingthe harmonic exact solution u from the projection errors |u − Fh,FEk u|H1 on the FE

spaces Vh,FEk , or |u−Fh,∇k u|H1 on the virtual harmonic spaces Vh,∆k . Recall that the FE

21

interpolant errors are available in [28, 29, 31, 32] for general functions in Sobolev spaces,whilst interpolant errors |u−Fh,∆k u|H1 are accessed in [19] for harmonic functions.

Theorem 5.1. Let T h = S be a family of polygonal partitions of Ω, Shk be the SBFEMspace based on T h, and Vh,FEk the FE spaces based on the conglomerate meshes Ph. Sup-

pose the same trace interpolant is used in the definitions of Πhk and Fh,∆k , and the exact

solution u ∈ H1 of the model problem (34) is sufficiently regular for them to make sense.If uhorΠh,S

k u ∈ Shk is the associated Galerkin SBFEM approximation, then

|u− uh|H1(Ω) ≤ |u−Fh,FEk u|H1(Ω). (37)

Proof. Firstly, we observe two orthogonality relations.

1. As for any Galerkin approximation, the SBFEM solution verify the orthogonalityproperty a(u − uh, v) = 0 ∀v ∈ Shk,0, which is paramount for error estimates forsuch methods.

2. Proposition 4.1 (i.e., Fh,FEk u ∈ Vh,FEk ⊂ Dhk), combined with properties (27) and(26), implies that

a(uh,Πhk u−F

h,FEk u) = 0. (38)

These two orthogonality relations imply the Pythagorean equality

|u−Fh,FEk u|2H1(Ω) = |u− uh|2H1 + |uh −Fh,FEk u|2H1(Ω).

Consequently, the estimate (37) holds.

Theorem 5.2. Let T h = S be a family of polygonal partitions of Ω, Shk and Vh,∆k bethe SBFEM and virtual spaces based on T h. Suppose the same trace interpolant is usedin the definitions of Πh

k and Fh,∆k , and the exact solution u ∈ H1 of the model problem(34) is sufficiently regular for them to make sense. If uh ∈ Shk is the associated GalerkinSBFEM approximation, then

|u− uh|H1(Ω) ≤ |u−Fh,∆k u|H1(Ω) + |Fh,∆k u− Πhk F

h,∆k u|H1(Ω). (39)

Proof. The result is a consequence of Galerkin orthogonality property

|u− uh|H1(Ω) = infv∈Shk|u− v|H1(Ω) ≤ |u− Πh

k u|H1(Ω),

the error decomposition (30), and the property Πhk u = Πh

k Fh,∆k u remarked in (32).

6. Numerical experiments

In this section, we present SBFEM simulation results for selected test problems. First,we consider problems with smooth solutions for the verification of the predicted theoreticalconvergence results of Section 5. For a two-dimensional problem, we explore discretiza-tions based on quadrilateral or polygonal S-elements subdivided into collapsed scaledtriangles. Then, a three-dimensional test problem is explored using SBFEM approxima-tions based on uniform hexahedral and polyhedral S-elements subdivided into collapsedscaled pyramids, and also on a more general geometry context of polyhedral S-elementssubdivided by scaled collapsed tetrahedra. For comparison, we present results obtained by

22

H1-conforming FE methods based on the meshes Ph formed by the agglomeration of thecorresponding triangles, pyramids, and tetrahedra partitions of the subdomains. Finally,we also evaluate the numerical performance of a coupled FEM+SBFEM formulation fora point singular problem, in which a traditional finite element formulation is modified bya scaled boundary element in the vicinity of the singularity.

For the current simulations, we implemented the method in the computational frame-work NeoPZ1, which is an open source finite element library whose objective is to facilitatethe development of innovative technology in finite elements [36]. Such a framework al-lows using a varied class of element geometries, applying mesh refinement, varying theapproximation order, and to approximate partial differential equations using differentapproximation spaces - H1, H(div), H(curl) and discontinuous -, as well as mixed andhybrid finite elements and multiscale simulations. As the NeoPZ was conceived usingobject-oriented concepts, with abstract classes, templates, and small blocks, it offeredthe required functionalities for a general coding of SBFEM simulations. Moreover, theconcept of element neighbours associated with geometric entities in NeoPZ was useful forthe construction of the collapsed geometric elements and definition of the scaled bound-ary partitions. Two and three dimensional SBFEM approximations applied to eitherLaplace’s equation or elasticity are implemented in a single class structure.

Example 6.1- smooth solution in 2D

The Laplace equation is approximated on the domain Ω = [−1; 1]× [−1; 1], where theharmonic problem (34) is considered with exact solution u(x, y) = exp (πx) sin (πy). Thisis the same problem of the interpolation example in Section 4.4, illustrated in Figure 5.

We approximate the problem by the Galerkin SBFEM using sequences of partitionsT h for three kinds of S-elements, with refinement levels h = 2−`, ` = 1, · · · 4: (i) uniformn× n, quadrilateral S-elements, n = 2`+1, each one having ΓS formed by its 4 edges, (ii)polygons with 8 edges obtained from uniform quadrilaterals whose sides are subdividedonce, and (iii) unstructured polygonal S-elements constructed using the mesh generatorsoftware PolyMesher [37], by giving as input the number of elements in x and y axes. Forthis sequence of four irregular polygonal partitions the scaled boundaries have averagecharacteristic width close to the adopt in the uniform contexts. Thus the same indexparameter h is adopted for them. Recall that each S-element is subdivided into trianglessharing the scaling center point as a vertex and having one facet in Γh,S as opposite edge.Figure 11 illustrates the particular partitions for h = 1

4.

Quadrilateral S-elements Polygonal S-elements - case 1 Polygonal S-elements - case 2

Figure 11: Example 6.1- Uniform quadrilateral and polygonal partitions T h, h = 14 : S-element distin-

guished by different colors and subdivided into scaled triangles.

1NeoPZ open-source platform: http://github.com/labmec/neopz

23

http://github.com/labmec/neopz

Table 1: Example 6.1- Galerkin SBFEM errors EhL2 = ‖u−uh‖L2(Ω) and Eh

H1 = |u−uh|H1(Ω) for uniform

partitions T h, h = 2−`, of quadrilateral and polygonal (case 1) S-elements.

Uniform quadrilateral S-elements

`k=1 k=2 k = 3

DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1

1 25 1.80E0 1.99E1 65 1.31E-1 2.56E0 105 7.78E-3 2.28E-12 81 4.50E-1 9.50E0 225 1.68E-2 5.92E-1 369 4.68E-4 2.62E-23 289 1.13E-1 4.68E0 833 2.12E-3 1.42E-1 1377 2.95E-5 3.19E-34 1089 2.82E-2 2.33E0 3201 2.65E-4 3.50E-2 5313 1.86E-6 3.96E-4

Rate 2.00 1.01 Rate 3.00 2.02 Rate 3.99 3.01

k = 4 k = 5 k = 6DOF Eh

L2 EhH1 DOF Eh

L2 EhH1 DOF Eh

L2 EhH1

1 145 5.87E-4 2.09E-2 185 3.89E-5 1.77E-3 225 1.96E-6 1.03E-42 513 1.99E-5 1.29E-3 657 6.04E-7 5.22E-5 801 1.53E-8 1.47E-63 1921 6.42E-7 8.14E-5 2465 9.47E-9 1.57E-6 3009 1.23E-10 2.21E-84 7425 2.03E-8 5.11E-6 9537 1.48E-10 4.81E-8 11649 9.46E-13 3.42E-10

Rate 4.98 3.99 Rate 6.00 5.03 Rate 6.99 6.02

Uniform polygonal S-elements - case 1

`k=1 k=2 k = 3

DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1

1 21 8.06E-1 1.23E1 45 8.77E-2 1.95E0 69 8.09E-3 2.53E-12 65 2.86E-1 6.66E0 145 1.55E-2 5.30E-1 225 4.98E-4 2.79E-23 225 1.54E-2 3.08E0 513 1.87E-3 1.22E-1 801 3.05E-5 3.24E-34 833 3.74E-3 1.50E0 1921 2.30E-4 2.97E-2 3009 1.92E-6 3.94E-4

Rate 2.08 1.04 Rate 3.02 2.04 Rate 3.99 3.02

k = 4 k = 5 k = 6DOF Eh

L2 EhH1 DOF Eh

L2 EhH1 DOF Eh

L2 EhH1


Rate 5.01 4.03 Rate 5.98 5.06 Rate 7.02 6.05

The energy and L2 errors summarized in Table 1 are for the Galerkin SBFEM solutionsin Shk based on uniform quadrilateral S-elements and uniform polygonal S-elements of case1, using polynomial orders 1 ≤ k ≤ 6. The numerical results are in accordance with thepredicted rates of order k for energy errors. Optimal rates of order k+1 are also observedfor the errors measured by the L2-norm.

In Figure 12, the energy and L2 errors are plotted versus the number of DOF forGalerkin SBFEM solutions in Shk based on the polygonal meshes of case 2. For comparison,the Galerkin FE solutions in Vh,FEk based on the associated scaled triangular partitionsPh are also shown, revealing comparable accuracy in both methods, but with less DOF inSBFEM simulations. Recall that SBFEM shape functions are determined by the tracesover scaled boundary elements, whilst FE spaces are also populated with shape functionsconnected with triangular DOF other than the edge ones opposed to the scaling center.One also observe that their error curves approach the possible optimal slopes −k and−(k+ 1) when measured by energy or L2 norms. This experiment illustrates the SBFEMflexibility with respect to mesh generation for numerical simulations without convergencedeterioration.

Plots illustrating SBFEM k-convergence histories in the energy norm versus DOF areshown in Figure 13, with k = 1, · · · , 6, and for S-elements with fixed boundary mesh sizeh = 1

4. The plots on the left are for the SBFEM interpolation in the single S-element

Ω (see Figure 5) and for the Galerkin SBFEM experiment for the uniform quadrilateralpartition T h of Figure 11. For both cases, the error decay as k increases shows a typical

24

Polygonal S-elements - case 2

Figure 12: Example 6.1- Energy and L2 errors versus DOF for the Galerkin SBFEM solutions in Shk ,based on the irregular polygonal S-elements of case 2 for the Galerkin FE solutions and for the GalerkinFE solutions in Vh,FE

k based on the associated scaled triangular partitions Ph, for k = 2, 4 and 6.

exponential convergence, but the interpolation experiment, by just refining the boundaryof a single element, requires less DOF for a given accuracy threshold. For comparison,k-convergence plots for two H1-conforming FE methods are also included: using Pk(K)polynomials in the triangles K of the conglomerate partitions Ph (FE), and for Duffy’sspaces Dhk,k(S) (Duffy’s FE). Errors using usual FE and collapsed FE are comparable, butthe latter has more equations to be solved. But what is more noticeable on these plots isthat SBFEM errors are not only smaller in magnitude than the FE errors (as predictedby Theorem 5.1, since energy FE errors are bounded by FE interpolation errors), butSBFEM requires less DOF to reach a given accuracy, the key property expected to beheld for operator adapted methods.

We also compare in Figure 13 (right) the k-convergence properties of the GalerkinSBFEM for spaces based on T h of the uniform quadrilateral and polygonal S-elements(case 1) of Figure 11. This comparison experiment shows that the use of the uniformpolygonal mesh of case 1 requires fewer equations to be solved for a given target error.On the other hand, a bigger eigenvalue system has to be solved for each S-element. Thiskind of polygonal mesh can be seen as a combination of refining both the boundary andinside the subdomains. Due to this flexibility, the SBFEM can generate octree (3D) orquadtree (2D) meshes [15, 38], giving high accuracy, without any additional techniques.

Quadrilateral S-elements Quadrilateral versus polygonal S-elements

Figure 13: Example 6.1- k-convergence histories versus the number of DOF, for k = 1, · · · , 6. Left:SBFEM interpolation Πh,S

k u based on the scaled partition of Figure 5, Galerkin SBFEM for Shk , Duffy’sFE for Dh

k,k, both based on the uniform quadrilateral partition T h of Figure 11, and FE method for

Vh,FEk based on the scaled triangular partition Ph. Right: Galerkin SBFEM solutions in Shk using the

uniform quadrilaterals and polygonal meshes of case 1 shown in Figure 11. All cases are for h = 14 .

25

Section 6.2: smooth solution in 3D

The second example refers to approximating Laplace’s equation on a 3D domain Ω =[0, 1]× [0, 1]× [0, 1], with exact harmonic solution

u(x, y, z) = 4(

exp(πx

4

)sin(πy

4

)+ exp

(πy4

)sin(πz

4

)).

This problem corresponds to the interpolation Example 2 of Section 4.4.Three types of geometry for T h are considered, each one with refinement levels h = 2−`,

` = 1, 2, and 3. The illustrations in Figure 14 are for h = 14. One is for n × n × n

uniform hexahedral partitions, n = 2`, where each S-element is decomposed into sixpyramids. The second one is composed of polygons (case 1) constructed by subdividingonce each square face of uniform hexahedral partitions into four uniform squares (for thisconfiguration, each S-element is a polyhedron with 24 quadrilateral facets, and composedby 24 scaled pyramids). More general polyhedral partitions (case 2) are constructed bythe software package Neper [39], by giving the number n of S-elements in x, y, and zdirections. Then, for each S ∈ T h, we applied gmsh [40] for the construction of theinternal tetrahedral partitions T h,S. The average edge characteristic sizes of the scaledboundary elements of these three irregular partitions resulted to be comparable to theparameter h of the uniform contexts. The pyramids and tetrahedra forming S are mappedby Duffy’s transformations from the reference hexahedron or prism, respectively.

Hexahedral S-elements Polygonal S-elements - case 1

Polygonal S-elements - case 2

Figure 14: Example 6.2- Hexahedral and polyhedral partitions T h, h = 14 : T h,S composed by scaled

pyramids (top), and by scaled tetrahedra (bottom).

The results for the Galerkin SBFEM solutions in Shk, with k = 1, · · · , 4, based on theuniform hexahedral S-elements and on the polyhedral S-elements of case 1 are documentedin Table 2. Optimal accuracy of order k for energy norm, and k+1 for the L2-norm occur.Energy and L2 errors obtained with the polyhedral partitions of case 2 are plotted versusDOF in Figure 15. For comparison, the Galerkin FE solutions in Vh,FEk based on theassociated scaled tetrahedral partitions Ph are also shown. Similar conclusions hold asfor the experiment shown in Figure 12. One can also observe that both Galerkin SBFEMand FE approximation errors have similar magnitude, but with less DOF in the SBFEM

26

Table 2: Example 6.2- Galerkin SBFEM errors EhL2 = ‖u − uh‖L2(Ω) and Eh

H1 = |u − uh|H1(Ω) for

uniform partitions T h of hexahedral and polyhedral (case 1) S-elements, with h = 2−`.

Uniform hexahedral S-elements

`k=1 k=2 k = 3 k = 4

DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1


Rate 2.03 1.02 Rate 2.96 2.00 Rate 3.97 3.01 Rate 4.98 4.00

Uniform polyhedral S-elements - case 1

`k=1 k=2 k = 3 k = 4

DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1



systems. Their error curves measured with energy and L2 norms also approach the possibleoptimal slopes −k and −(k + 1), respectively.

Polyhedral S-elements - case 2

Figure 15: Example 6.2- Energy and L2 errors versus DOF for the Galerkin SBFEM solution in Shk , fork = 1, · · · , 4, based on the irregular polyhedral S-elements of case 2.

Hexahedral S-elements Hexahedral vs. Polyhedral S-elements

Figure 16: Example 6.2- k-convergence histories as function of the number of DOF, for k = 1, · · · , 4:Left: SBFEM interpolation Πh,S

k u for the scaled partition T h,S of Figure 7, Galerkin SBFEM for Shkbased on uniform hexahedral partition T h, and FE method for Vh,FE

k based on the conglomerated scaledpyramidal partition Ph. Right: Galerkin SBFEM for Shk based on hexahedral and polyhedral S-elementsof case 1. In all the experiments, h = 1

4 .

In the left side of Figure 16, we compare the SBFEM k-convergence using the fixed

27

uniform hexahedral partition at the refinement level h = 14, shown in Figure 14, with

equivalent results for the FE method using the spaces Vh,FEk ⊂ H1(Ω) based on theassociated pyramidal partition Ph. SBFEM approximations lead to lower error values, aspredicted by Theorem 5.1, and the linear systems have a reduced number of equations.The error curve of the interpolation experiment illustrated in Figure 8 is also included.

The plots on the right side compare the the k-convergence of the two SBFEM solutionsin Shk based on the uniform hexahedral partition and on the polyhedral partition of case1 illustrated in Figure 14, both with h = 1

4. Similarly to the comparison experiment of

the previous example, shown in Figure 13, these convergence histories also show that theuse of polygonal mesh requires fewer equations to be solved for a given target error, butreminding that it requires bigger eigenvalue systems to be solved for the computatiobn ofSBFEM shape functions in the S-elements.

Section 6.3: coupled FE-SBFEM formulation for a singular problem

Taking the singular harmonic function interpolated in Section 4.4, namely

u = 21/4√r cos(

θ

2) = 2−1/4

√x+

√x2 + y2,

we enforce Dirichlet boundary condition on (x, 0), x < 0, and Neumann boundary con-

dition elsewhere. Due to the lack of regularity of u ∈ H32−ε(Ω), the error estimates of

Theorem 5.1 in terms of FE interpolant error based on regular partitions are restrictedin theory to order h

12−ε. This problem was considered in [41] to evaluate the efficiency of

the mixed FE method when quarter-point elements are used in the vicinity of the originO = (0, 0) (singular point), showing dramatic accuracy improvement. Recall that thespecific 6-noded quarter-point element is also of Duffy’s type, obtained by collapsing areference quadrilateral element on triangles.

h = 12 h = 1

16

Figure 17: Example 6.3- Meshes for the coupled FE-SBFEM formulation: FE (blue) in the smooth regionand SBFEM (magenta) close to the singularity point.

With this motivation, we propose a formulation composing SBFEM approximations ina single element S = [−0.5, 0.5]× [0, 0.5] and FE approximations elsewhere, in the regionwhere the solution is smooth. Similarly to the interpolation experiments in Section 4.4, thespace Shk(S) is conceived in such a way that the scaling center is located on the singularitypoint, which means that an open scaled boundary element is applied. The vertical andtop-horizontal edges of S are uniformly subdivided to form an interface partition Γh,S.Elsewhere, a uniform quadrilateral mesh matching Γh,S is adopted, as illustrated in Fig17 for h = 1

2and h = 1

16. The coupling between FE and SBFEM approximations is

straightforward since SBFEM uses compatible FE spaces at the interface. Four meshsizes h = 2−`, ` = 1, · · · , 4, and polynomials of degree k = 1, · · · , 4 are performed.

28

The corresponding results are documented in Table 3. As for regular problems withsmooth solutions, optimal rates of convergence of order k and k + 1 for energy and L2

errors hold for this singular problem, without any adaptivity, i.e. uniform degree k is usedover the domain and no h-adaptivity is applied as well.

Table 3: Example 6.3- Errors EhL2 = ‖u− uh‖L2(Ω) and Eh

H1 = |u− uh|H1(Ω), h = 2−`, for the combinedFE-SBFEM method.

`k=1 k=2 k = 3 k = 4

DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1 DOF EhL2 Eh

H1

1 14 8.44E-4 1.11E-1 26 7.87E-4 1.94E-2 76 9.17E-5 2.82E-3 125 1.19E-5 5.25E-42 39 2.02E-3 5.54E-2 117 1.12E-4 4.23E-3 259 6.35E-6 3.83E-4 441 4.59E-7 3.72E-53 125 4.95E-4 2.73E-2 665 1.45E-5 1.06E-3 949 4.16E-7 4.87E-5 1649 1.54E-8 2.39E-64 441 1.23E-4 1.36E-2 4401 1.84E-6 2.66E-4 3625 2.67E-8 6.10E-6 6369 4.97E-10 1.50E-7


For comparison, two k-convergence histories as function of the number of DOF areshown in Figure 18 for fixed partitions of the domain Ω: one for the SBFEM interpolationerrors computed in Example 3, Section 4.4, and the other for the combined GalerkinFE-SBFEM method. The partitions used in these experiments are illustrated in Figure18, noticing that they coincide within the region S around the singularity, but the FEpartition in the smooth region being more refined. Whilst SBFEM interpolation in thesingle element Ω requires much less DOF, both experiments reach very close error values,because the error in this problem is governed by the singularity, modeled using SBFEMin both experiments. However, the results of Fig 18 could be deceiving. It should beemphasized that global SBFEM interpolation in the whole domain Ω was feasible in thisparticular test problem, but this will not the case in practical singular problems, for whichcoupled FE+SBFEM simulations reveal to be a simple and efficient option.

SBFEM interpolation

FE-SBFEM

Figure 18: Example 6.3- Partitions and k-convergence histories versus the number of DOF, with k =1, · · · , 4, for SBFEM interpolants Πh,S u of Example 3, Section 4.4, and Galerkin FE-SBFEM solutions.

7. Conclusions

We provide a priori error estimates in energy norm for Galerkin SBFEM approxima-tions of harmonic solutions by exploring two aspects of SBFEM’s methodology.

The SBFEM approximation spaces are based on star-shaped polytopal subregions (S-elements), where the functions are parametrized in the radial and surface directions. Weshow that they can be presented in the context of Duffy’s approximations based on sub-partitions of the S-elements. Piecewise polynomial discretization is adopted for surface

29

traces, which are radially extended to the interior of S by solving local harmonic problemsusing test functions restricted to Duffy’s spaces. As a consequence, shape functions canbe derived from analytical solutions defined by eigenvalue problems, whose coefficientsare determined by the geometry of the S-elements.

We demonstrate that there is an equivalence between the SBFEM ODE equation andan orthogonality property of SBFEM spaces, with respect to the gradient inner productfor a wide class of Duffy’s approximations vanishing on the facets of S. This orthogonalproperty is the key for the derivation of the estimation of SBFEM errors in energy norm.The Galerkin SBFEM approximation error is necessarily smaller than the FE interpolanterror for the FE space included in the Duffy’s space sharing the same interface traces.

We show that SBFEM errors in the approximation of harmonic functions come fromtwo sources: there is the kind of error caused when the trace of harmonic functions arediscretized over the facets of S, occurring in virtual harmonic approximations, and thereis the error caused by the deviation of SBFEM approximations of being harmonic insideS. The fact that the first source of error is the dominant one is favorable for applicationsfor singular problems, where the singularity may be isolated, without interaction with theS-element facets. For this class of problems, the solution away from the singularity isregular. If the convergence rate is dominated by the approximation on the boundary ofS, then this explains regular convergence rates even for singular problems, as illustratedby the verification simulations.

Numerical tests in 2D and 3D problems emphasize the optimal rate of convergence ofthe scaled boundary approximations, proven theoretically for the energy norm. Althoughwe have considered only harmonic solutions, the demonstration can be extended for moregeneral homogeneous elliptic PDEs, for instance, elasticity problems without body loads.

Acknowledgements

The authors thankfully acknowledge financial support from: FAPESP - Sao Paulo Re-search Foundation, grants 2016/05155-0 (Gomes) and 17/08683-0 (Devloo), CNPq - Con-selho Nacional de Desenvolvimento Cientıfico e Tecnologico, grants 305823-2017-5 (De-vloo) and 306167/2017-4 (Gomes), and ANP - Brazilian National Agency of Petroleum,Natural Gas and Biofuels, grant 2014/00090-2 (Coelho, Devloo).

References

[1] C. Song, J. P. Wolf, The scaled boundary finite-element method—alias consistentinfinitesimal finite-element cell method—for elastodynamics, Computer Methods inApplied Mechanics and Engineering 147 (1997) 329–355.

[2] C. Song, J. P. Wolf, The scaled boundary finite-element method: analytical solutionin frequency domain, Computer Methods in Applied Mechanics and Engineering 164(1998) 249–264.

[3] J. P. Wolf, The Scaled Boundary Finite Element Method, John Wiley & Sons, 2003.

[4] C. Song, The Scaled Boundary Finite Element Method: Theory and Implementation,John Wiley & Sons, 2018.

[5] I. Babuska, J. M. Melenk, The partition of unity method, International Journal forNumerical Methods in Engineering 40 (1997) 727–758.

30

[6] J. M. Melenk, Operator Adapted Spectral Element Methods I: Harmonic and Gen-eralized Harmonic Polynomials, Numerische Mathematik 1 (1999) 35–69.

[7] Z. Yang, Fully automatic modelling of mixed-mode crack propagation using scaledboundary finite element method, Engineering Fracture Mechanics 73 (2006) 1711–1731.

[8] C. Song, E. T. Ooi, S. Natarajan, A review of the scaled boundary finite elementmethod for two-dimensional linear elastic fracture mechanics, Engineering FractureMechanics 187 (2018) 45–73.

[9] A. L. N. Pramod, R. K. Annabattula, E. T. Ooi, C. Song, S. Natarajan, Others,Adaptive phase-field modeling of brittle fracture using the scaled boundary finiteelement method, Computer Methods in Applied Mechanics and Engineering 355(2019) 284–307.

[10] H. Guo, E. T. Ooi, A. A. Saputra, Z. Yang, S. Natarajan, E. H. Ooi, C. Song,A quadtree-polygon-based scaled boundary finite element method for image-basedmesoscale fracture modelling in concrete, Engineering Fracture Mechanics 211 (2019)420–441.

[11] J. Bulling, H. Gravenkamp, C. Birk, A high-order finite element technique withautomatic treatment of stress singularities by semi-analytical enrichment, ComputerMethods in Applied Mechanics and Engineering 355 (2019) 135–156.

[12] Y. Liu, A. A. Saputra, J. Wang, F. Tin-Loi, C. Song, Automatic polyhedral meshgeneration and scaled boundary finite element analysis of STL models, ComputerMethods in Applied Mechanics and Engineering 313 (2017) 106–132.

[13] S. Natarajan, P. Dharmadhikari, R. K. Annabattula, J. Zhang, E. T. Ooi, C. Song,Extension of the scaled boundary finite element method to treat implicitly definedinterfaces without enrichment, Computers & Structures 229 (2020) 106–159.

[14] A. Saputra, H. Talebi, D. Tran, C. Birk, C. Song, Automatic image-based stressanalysis by the scaled boundary finite element method, International Journal forNumerical Methods in Engineering 109 (2017) 697–738.

[15] K. Chen, D. Zou, X. Kong, X. Yu, An efficient nonlinear octree SBFEM and itsapplication to complicated geotechnical structures, Computers and Geotechnics 96(2018) 226–245.

[16] H. Gravenkamp, A. A. Saputra, S. Duczek, High-order shape functions in the scaledboundary finite element method revisited, Archives of Computational Methods inEngineering (2019) 1–22.

[17] H. Gravenkamp, A. A. Saputra, S. Eisentrager, Three-dimensional image-based mod-eling by combining SBFEM and transfinite element shape functions, ComputationalMechanics (2020) 1–20.

[18] M. G. Duffy, Quadrature over a pyramid or cube of integrands with a singularity ata vertex, SIAM Journal on Numerical Analysis 19 (1982) 1260–1262.

31

[19] A. Chernov, L. Mascotto, The harmonic virtual element method: stabilization andexponential convergence for the Laplace problem on polygonal domains, IMA Journalof Numerical Analysis, 39(4), 1787-1817 33 (2019) 1787–1817.

[20] J. N. Lyness, R. Cools, A survey of numerical cubature over triangles, TechnicalReport, Argonne National Laboratory, 1994.

[21] M. G. Blyth, C. Pozrikidis, A Lobatto interpolation grid over triangle, IMA Journalof Applied Mathematics 71 (2006) 153–169.

[22] G. Karniadakis, S. Sherwin, Spectral/hp Element Methods for Computational FluidDynamics, Oxford University Press, 1999.

[23] Y.-L. Wu, Collapsed isoparametric element as a singular element for a crack normalto the bi-material interface, Computers & Structures 47 (1993) 939–943.

[24] S. L. Pu, M. Hussain, W. E. Lorensen, The collapsed cubic isoparametric element asa ingular element for crack probblems, International Journal for Numerical Methodsin Engineering 12 (1978) 1727–1742.

[25] I. S. Raju, Calculation of strain-energy release rates with higher order and singularfinite elements, Engineering Fracture Mechanics 28 (1987) 251–274.

[26] J. Shen, L.-L. Wang, H. Li, A Triangular Spectral Element Method Using FullyTensorial Rational Basis Functions, SIAM Journal on Numerical Analysis 47 (2008)1619–1650.

[27] N. Nigam, J. Phillips, High-order conforming finite elements on pyramids, IMAJournal of Numerical Analysis 32 (2012) 448–483.

[28] I. Babuska, M. Suri, The h-p version of the finite element method with quasiuniformmeshes, RAIRO - Modelisation Mathematique et Analyse Numerique 21 (1987) 199–238.

[29] R. Munoz-Sola, Polynomial lifting on the tetrahedron and applications to the h-pversion of the finite element method in three dimensions, SIAM Journal on NumericalAnalysis 34 (1997) 282–314.

[30] L. Demkowicz, Polynomial Exact Sequences and Projection-Based Interpolationwith Application to Maxwell Equations, in: D. Boffi, L. Gastaldi (Eds.), Mixed Fi-nite Elements, Compatibility Conditions, and Applications, Lecture Notes in Math-ematics, Vol. 1939, Springer, 2008, pp. 101–158.

[31] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz, A. Zdunek, Com-puting with hp-Adaptive Finite Elements, Vol. 2, Chapman and Hall, 2007.

[32] J. M. Melenk, C. Rojik, On commuting p-version projection-based interpolation ontetrahedra, Mathematics of Computation 89 (2020) 45–87.

[33] M. Bergot, G. Cohen, M. Durifle, Higher-order finite elements for hybrid meshesusing new nodal pyramidal elements, Journal of Scientific Computing 42 (2010)345–381.

32

[34] C. Schwab, p-and hp-Finite Element Methods: Theory and Applications in Solid andFluid Mechanics, Clarendon Press Oxford, 1998.

[35] A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Vol. 159, SpringerScience & Business Media, 2013.

[36] P. R. B. Devloo, PZ: An object oriented environment for scientific programming,Computer Methods in Applied Mechanics and Engineering 150 (1997) 133–153.

[37] C. Talischi, G. H. Paulino, A. Pereira, I. F. M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Structural andMultidisciplinary Optimization 45 (2012) 309–328.

[38] A. A. Saputra, S. Eisentrager, H. Gravenkamp, C. Song, Three-dimensional image-based numerical homogenisation using octree meshes, Computers & Structures 237(2020) 106263.

[39] R. Quey, P. R. Dawson, F. Barbe, Large-scale 3D random polycrystals for the finiteelement method: Generation, meshing and remeshing, Computer Methods in AppliedMechanics and Engineering 200 (2011) 1729–1745.

[40] C. Geuzaine, J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, International Journal for Numerical Methods inEngineering 79 (2009) 1309–1331.

[41] D. de Siqueira, A. M. Farias, P. R. B. Devloo, S. M. Gomes, Mixed finite elementapproximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements, Applied Numerical Mathematics 158 (2020)85–102.

Appendix A. Scaled ODE equation

The second-order ODE problem (15) can be solved using standard methods through a

system of first-order differential equations. Given Qi(ξ) =

[ξd−1E

11Φ′

i(ξ) + ξd−2E21

Φi(ξ)],

the ODE (15) can be expressed by the two equations:

ξΦ′i(ξ) =

(−E−1

11E

12+ 0.5(d− 2)I

)Φi(ξ) + E−1

11Qi(ξ), (A.1)

ξQ′i(ξ) =

(−E

21E−1

11E

12+ E

22

)Φi(ξ) +

(E

11E−1

21− 0.5(d− 2)I

)Qi(ξ). (A.2)

This ODE system can be grouped in a matrix form as

ξX ′(ξ) = −Z X(ξ), ξ ∈ [−1, 1], (A.3)

for X(ξ) =

[Φ(ξ)

Q(ξ)

], where Φ(ξ) = [Φi(ξ)], and Q(ξ) = [Q

i(ξ)] are N S × N S matrices with

columns Φi(ξ) and Qi(ξ), and Z is the 2N S × 2N S matrix

Z =

(E−111E

12− 0.5(d− 2)I

)−E−1

11

−E22

+ E21E−1

11E

21

(−E

21E−1

11+ 0.5(d− 2)I

) .

33

If

[A

Q

]are linearly independent eigenvectors of the matrix Z corresponding to eigenvalues λ,

then the function X(ξ) =

[A

Q

]ξλ solves (A.3). The functions ξλ corresponding to eigenvalues

having negative real parts are unbounded for ξ → 0, and are unsuited to describe solutions atthe interior of the S-element, whilst those of positive real parts represent solutions that are zeroat the scaling center of S. Thus, the desired solutions of the system (A.1)-(A.2) are taken as

Φ(ξ) = A+

diag(ξλ+), Q(ξ) = Q+

diag(ξλ+),

where λ+ ∈ RN represents the positive real part of λ, A+

= [A+i] and Q+

= [Q+i

] are the

associated eigenvector components. For simplicity, the index + is dropped in Section 3.3.

34

Error estimates for the Scaled Boundary Finite Element Method

Documents

Transcript of Error estimates for the Scaled Boundary Finite Element Method