Cut Finite Element Methods on Parametric Multipatch Surfaces

Cut Finite Element Methods onParametric Multipatch Surfaces

Tobias Jonsson

Licentiate ThesisDepartment of Mathematics and Mathematical StatisticsUmea UniversitySE-901 87, Umea, Sweden

List of papersThis thesis consists of an introduction to the area and following papers:

Paper I. T. Jonsson, M. G. Larson, and K. Larsson. Cut finite elementmethods for elliptic problems on multipatch parametric surfaces. Comput.Methods Appl. Mech. Engrg., 324:366–394, 2017.

Paper II. P. Hansbo, T. Jonsson, M. G. Larson, and K. Larsson. ANitsche method for elliptic problems on composite surfaces. Comput.Methods Appl. Mech. Engrg., 326:505–525, 2017.

Paper III. T. Jonsson, M. G. Larson, and K. Larsson. Graded Para-metric CutFEM and CutIGA for Elliptic Boundary Value Problems inDomains with Corners. arXiv e-prints, Dec. 2018.

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Acknowledgments

First of all I would like to thank my main advisor Professor Mats G.Larson for introducing me to the world of finite elements and providingme with his great knowledge and ideas during the work of this thesis.

I also want to give sincere gratitude towards my assistant advisor DrKarl Larsson for his support, encouragement, and always having time forme.

Finally, I would like to thank Linnea and my family for all the loveand support during this thesis.

Tobias JonssonUmea, January 2019

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Contents

1 Introduction 11.1 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Cut Finite Element Method 42.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Finite Element Approximation . . . . . . . . . . . . . . . . 72.3 The Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Nitsche’s Method . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Partial Differential Equations on Surfaces 123.1 Surface Problems . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Patches and CAD . . . . . . . . . . . . . . . . . . . . . . . 143.3 Parametric Description . . . . . . . . . . . . . . . . . . . . . 16

4 Summary of Papers 20

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Chapter 1

Introduction

Partial differential equations are widely used to model real-world phenom-ena appearing in our everyday life. Examples are fluid flows, elasticity,heat diffusion, and many more. Typically it is not possible to find anexact solution to these problems, instead we rely on numerical methodsto find approximated solutions. Since the solution is not exact, estimatesof both the error magnitude and convergence rate is of high importance.

In this thesis we develop new Cut Finite Element methods (CutFEM)which is a framework for finite element methods (FEM). The main ideain FEM is to partition the computational domain into simpler geometricobjects, called elements. They can be triangles, quadrilaterals, or tetra-hedrons depending on the dimension. In contrast CutFEM allows thephysical domain to cut through the grid of elements, which simplifies themesh generation. In engineering the mesh creation is a time consumingand expensive task. There is an increasing interest for robust methods todeal with complex and/or evolving geometries where the standard meth-ods are too cumbersome and expensive.

Traditionally, numerical methods have focused on techniques for ob-taining approximations of the physical equations rather than the geometry.We choose to work with problems on surfaces, since they require accurategeometric description. Some applications are the modeling of lubrication,and stresses on elastic membranes.

A common approach to deal with surface geometry is by using computer-aided design programs (CAD software), which divides the surface intoseveral trimmed patches. Each patch has a local representation and a

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transformation map from a reference domain onto the surface. An il-lustration of this parametric setting is the Earth’s surface which can berepresented by two angular coordinates (parameters) longitude and lati-tude. The land mass of Europe could be one patch, which would have acorresponding flat representation in a reference domain.

The outline of this thesis is as follows: In Section 2 we present thebackground and theory for the cut finite element method; in Section 3the parametric setting is described and we show how to use it in weakformulation; and finally in Section 4 we give a brief summary of eachpaper.

1.1 Thesis objectives

The main objectives for this thesis have been:

• Develop methods which combines the parametric surface settingwidely used in CAD together with the Cut Finite Element method.

• Implement and provide numerical examples.

• Derive a priori error estimates for the given methods.

1.2 Main results

• Developed a method for computations on multipatch parametric sur-faces which is based on a fictitious domain method (CutFEM). Thisapproach does not require the construction of conforming meshes forthe trimmed patches. (Paper I)

• By adding a stabilization term, which extends the coerciveness to cutfinite elements spaces, we can perform the standard error analysisindependent of how the trim curves cut the mesh. (Paper I)

• We have developed a quadrature routine for integrating cut elementstrimmed by a piecewise smooth curves. (Paper I)

• A new Nitsche method for a diffusion problem which can handle in-terfaces where more than two surfaces meet. The method is derived

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based on a Kirchhoff condition where the sum of all co-normal fluxesare zero at the interface. (Paper II)

• We have developed a cut finite element method for elliptic prob-lems with nonconvex corner singularities. From known informationabout the singularity a suitable map was created that grades thecomputational mesh towards the singularity. (Paper III)

• Proved a priori error estimates by using bounds on the derivativesin the reference domain in terms of a weighted norm in the physicaldomain. (Paper III)

Keywords: cut finite element method, Nitsche method, a priori errorestimation, parametric map, interface problem

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Chapter 2

The Cut Finite ElementMethod

Standard Finite Element Method. The main idea behind the finiteelement method (FEM) is to discretize the computational domain intosimpler geometric objects, called elements. They can be triangles, quadri-laterals, or tetrahedrons depending on the dimension. A set of basis func-tions, typically polynomials, is derived from the elements to approximatethe solution. We derive the method by rewriting the problem in weakform, which is obtained by multiplying with a test function and integrat-ing by parts over the domain. Solution is then found by using variationalmethods in a finite dimensional subspace Vh of the infinite Hilbert space Vand we are left to solve a linear system of equations. We refer to [2,3,11,14]for further reading.

Cut Finite Element Method. In engineering the mesh creation is atime consuming and expensive task. There is an increasing interest forrobust methods to deal with complex and/or evolving geometries. Themain motivation for the Cut finite element method (CutFEM) is to makethe mesh discretization less dependent of the underlying geometry con-straints [5]. In this approach the geometry is allowed to arbitrarily cutthrough the elements. We embed the domain of interest in a fixed back-ground mesh with simple structure. Elements cut by the boundary couldbe arbitrarily small and cause badly conditioned system matrix and somestabilization is needed.

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2.1 Model Problem

We would like to show the different approaches by considering the follow-ing problem: find u such that

−∆u = f in Ω (2.1)

u = g on ∂Ω (2.2)

where Ω is a bounded domain in R2 which is shown in Figure 2.1, with itsboundary ∂Ω, ∆ = ∂2

∂x2+ ∂2

∂y2is the Laplacian and f is a given function.

Figure 2.1: The computational domain Ω and its boundary ∂Ω.

Weak Form. To solve the equation we start by rewriting the probleminto its weak form, which is a variational equation. The weak formulationis useful for solving partial differential equations, since it holds in an av-erage sense over the domain instead of absolutely satisfy the equation ineach point.

The weak form is obtained by multiplying both sides of (2.1) with atest function v with v = g1 on ∂Ω and integrating by parts over Ω.

1Here the boundary condition is strongly imposed, later we will present a methodto weakly impose it.

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∫

Ωfv dx = −

∫

Ω∆uv dx (2.3)

=

∫

Ω∇u · ∇v dx−

∫

∂Ωn · ∇uv ds (2.4)

=

∫

Ω∇u · ∇v dx (2.5)

by assuming v = 0 on ∂Ω. For the integrals above to exist we demand thatboth v and its gradient ∇v to behave nicely. This leads us to introducingthe Sobolev spaces of order k with their corresponding inner product as

(v, w)Hk(Ω) =∑

|α|≤k

(Dαv,Dαw)L2(Ω), ‖v‖2Hk(Ω) = (v, v)Hk(Ω) (2.6)

where L2(Ω) = H0(Ω) and (v, w)L2(Ω) =∫

Ω vw dx is the L2(Ω) innerproduct. By considering test functions in a subset of H1(Ω) we gaincontrol over both the function v and its gradient ∇v. The appropriatetest space V0 and trial space Vg are thus

V0 = v ∈ H1(Ω) : v = 0 on ∂Ω (2.7)

Vg = v ∈ H1(Ω) : v = g on ∂Ω (2.8)

By defining the bilinear form a(u, v) = (∇u,∇v)L2(Ω) and the linear func-tional l(v) = (f, v)L2(Ω) we can write the problem in weak form: Given afunction f , find u ∈ Vg such that

a(u, v) = l(v) ∀v ∈ V0 (2.9)

This setting can be used to analyse a larger class of similar problems withthe same techniques.

Lax-Milgram Lemma. It is important to show that the solution of agiven problem both exists and is unique. This can be done using Lax-Milgram lemma if the abstract forms a(u, v), l(v) are continuous and thebilinear form a(u, v) also is coercive. Those requirements can be writtenas

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|a(u, v)| . ‖u‖Hs ‖v‖Hs (Continuous) (2.10)

a(v, v) & ‖v‖2Hs (Coercive) (2.11)

|l(v)| . ‖v‖Hs (Continuous) (2.12)

where s = 1 for the model problem and a . b means that there exists apositive constant C such that a ≤ Cb.

2.2 Finite Element Approximation

In general it is not possible to find a analytical solution u to (2.1) andwe thus tries seek a solution uh in a finite subspace Vh ⊂ V . The spaceis typically chosen to consist of piecewise polynomials of a given order p,since they are easy to manipulate. By replacing the space V with Vh, inweak form (2.9), the finite element method reads: find uh ∈ Vh such that

a(uh, v) = l(v) ∀v ∈ Vh (2.13)

The finite subspace Vh is spanned by a set of N linearly independent basisfunctions φi(x)Ni=1 which we can form the approximate solution as thelinear combination:

uh =

N∑

i=1

ξiφi(x) (2.14)

where ξi is the unknown coefficients and represent the degrees of free-dom for the solution uh. Inserting the linear combination of uh in thevariational form give us a N ×N linear system of equations as:

N∑

i=1

a(φi, φj)ξi = l(φj) for j = 1, . . . , N (2.15)

where the test function v is chosen to be φj(x). This can also be writtenwith matrices as

AU = F (2.16)

with components Aij = a(φi, φj), Ui = ξi, and Fj = l(φj).

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(a) A standard trianglemesh for the domain Ω.

(b) A quadrilateral cut meshfor the domain Ω.

Figure 2.2

2.3 The Mesh

To obtain a finite discretization we immerse the domain in a backgroundmesh, which is chosen to be structured and simple to work with. A com-mon choice in 2 dimensions is quadrilaterals and we show the differencesbetween a cut mesh and a regular mesh in Figure 2.2. We let Kh,0 bea quasi-uniform structured mesh consisting of elements K with mesh pa-rameter 0 < h ≤ h0. This partition of Ω is also where the approximatesolution uh is represented. We denote Kh as the elements in the back-ground mesh which intersects with Ω. This restriction is called the activemesh and is written as

Kh = K ∈ Kh,0 : K ∩ Ω 6= ∅ (2.17)

where only the elements affecting the geometry are remaining. Most ofthe time it will include cut elements in the vicinity of the boundary. Weillustrate the active and background mesh in Figure 2.3a.

2.4 Nitsche’s Method

The condition in (2.1) is called the Dirichlet boundary condition and isusually point-wise imposed. This is done by seeking the solution u in asolution space Vg which only includes functions that satisfy u = g on theboundary.

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In this thesis we will instead use Nitsche’s method [15] to imposeDirichlet boundary conditions weakly. The condition u = g on ∂Ω isenforced weakly by penalizing the jump u− g as

β(µ(u− g), v)∂Ω = β

∫

∂Ωµ(u− g)v ds (2.18)

where β > 0 is a penalty parameter and µ acts as a weight function.Nitsche proved that a good choice is µ ∝ h−1, where h is the mesh size.This is added to the system a(u, v) = l(v), but the weak form still lackssymmetry due to the consistency term (n · ∇u, v)L2(∂Ω). To retrieve sym-metry (n · ∇v, (u− g))L2(∂Ω) is added and the final weak form is

a(u, v) = (∇u,∇v)L2(Ω) − (n · ∇u, v)L2(∂Ω) − (n · ∇v, u)L2(∂Ω)

+ βh−1(u, v)L2(∂Ω) (2.19)

l(v) = (f, v)L2(Ω) − (n · ∇v, g)L2(Ω) + βh−1(g, v)L2(Ω)

Interface Conditions. By the same merits it is possible to imposeinterface conditions with Nitsche’s method [9].

2.5 Stabilization

Cut elements can be arbitrary small which causes numerical instability inthe solution. This is manifested in a very high condition number and lossof accuracy for the finite element solution uh. By adding a stabilizationterm called Ghost penalty we regain the control over elements that arebeing partially ”outside” of the domain [4]. This recovers the properties ofstandard finite element methods by extending the coercivity to the wholedomain. The stabilization added takes the form

sh(v, w) =∑

F∈Fh

sh,F (v, w), sh,F =

p∑

k=1

γk

([Dk

nv], [Dknw])F

(2.20)

where Fh is the set of interior faces F in the active mesh Kh such thatits element is cut by the boundary ∂Ω, γk > 0 is a positive constant and

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Kh,0

Kh

1

(a) The background grid Kh,0 with theactive mesh Kh in yellow.

Fh

1

(b) The set of interior faces Fh shownin red.

Figure 2.3

[Dknv] is the jump in the face normal derivative of order k. The stabilized

bilinear form then have the form

Ah(v, w) = ah(v, w) + sh(v, w) (2.21)

and we illustrate the set of interior faces Fh in Figure 2.3b.

2.6 Error Estimation

By computing the finite element solution uh we only obtain an approx-imation of the exact solution u. It is therefore necessary to analyze theerror e = u − uh to quantify the quality and accuracy of uh. There aretwo types of error estimates, namely a priori error and a posteriori error.From the former estimate we obtain an upper bound for the error in termsof the solution u and usually the mesh size parameter h. The estimatesare of interest since they can be used to find the order of convergenceof the specific finite element method. By convergence we mean that theerror u − uh in a given norm goes to zero as the mesh size h get refined.Usually the dependence is of polynomial order p for the mesh size h, i.eof the form:

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‖u− uh‖ ≤ chp (2.22)

for some norm ‖ · ‖ and c = c(Ω) is a constant. We will derive and usea priori error estimates in this thesis to test and verify our methods.

The latter estimate give instead an upper bound in terms of the nu-merical solution uh and the mesh size h. The posteriori error estimate isuseful when dealing with adaptive finite element methods. In this thesiswe will not use posteriori error estimates and we refer to [1,8] for furtherreading.

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Chapter 3

Partial DifferentialEquations on Surfaces

Partial differential equations on surfaces arises in several applications suchas transport phenomena and elastic membranes. Surfaces are usually rep-resented in two ways, either explicitly or implicitly. In this thesis we willuse the former surface description, also known as parametric. Below wegive a brief overview between the two approaches.

Explicit A surface, or more precise a two-dimensional manifold, can bethought of as a subset of R3 which locally looks like R2. The surface ofEarth is a sphere but appears to be flat for an observer standing on theground. This motivating example gives an idea of how we could representsurfaces as subsets of Euclidean space. Each point on a surface has aneighbourhood which is homemorphic to an open subset of R2 which wecall the reference surface. In other words there exists a bijective continuousmap between a flat reference surface ⊂ R2 and a possible curved surface⊂ R3. In general there exist no global map for the whole domain, insteadthe manifold is divided into patches (also called charts), that may overlap.The collection of patches and their corresponding bijection functions arecalled an Atlas. To describe the patches explicitly we require functionsof two variables since the map is between two surfaces. This explicitframework can be modelled such that each surface coordinates are givenby a continuous functions, e.g x = g1(s, t), y = g2(s, t), z = g3(s, t).

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Implicit An implicit surface is defined as all coordinates (x, y, z) whichsolves F (x, y, z) = 0. It is to be noted that it is not possible, most ofthe times, to solve for the coordinates explicitly. To give an example, theimplicit equation for the sphere is given by

x2 + y2 + z2 −R2 = 0 (3.1)

where R is the radius. Due to the lack of explicit coordinate representationthere are no direct ways to generate the surface points. However we knowfrom F (x, y, z) = 0 whether or not a point is on the surface. The implicitequation is also called the level set function and is usually modelled suchthat

F (x, y, z) > 0, above/inside the surface (3.2)

F (x, y, z) = 0, on the surface (3.3)

F (x, y, z) < 0, below/outside the surface (3.4)

3.1 Surface Problems

Here we list some real world problems that arise on surfaces.

Poisson Problem. The Poisson equation could be used to model thebehaviour of electric and fluid potentials, and the heat distribution for thestationary heat equation. On surfaces the Laplace operator ∆ is replacedwith the Laplace-Beltrami operator ∆Ω which is defined on Riemannianmanifolds. Let Ω be a smooth connected surface immersed in Rd andf ∈ C2(Ω), then the Laplace-Beltrami problem is: find u

−∆Ωu = f in Ω (3.5)

where ∆Ω = ∇Ω · ∇Ωu and ∇Ω the tangential gradient operator. Byadding a time derivative we obtain Linear diffusion on a surface:

∂u

∂t−D∆Ωu = f in Ω (3.6)

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where D is a diffusion constant. The diffusion equation models the randommovement of particles in a material due to the particles stochastic nature.By adding the physical phenomena advection to the diffusion equation weobtain the advection-diffusion equation:

∂u

∂t−D∆Ωu+∇Ω · (vu) = f in Ω (3.7)

where v is the velocity field of the quantity u. Advection is the bulkmovement of the regarded substance, and an example would be somematerial dissolved in the fluid with water flow v. The advection-diffusionequation can be used to model the transport of surfactants along fluid-fluid interfaces, grain boundary motion due to diffusion, and dealloyingmetals from surface dissolution [7].

Cahn-Hilliard Equation. The Cain-Hilliard equation is a fourth orderPDE which describes the phase separation process of two fluid compo-nents. Applications can be found in areas such as planet formation, foammodeling, interfacial flows, and phase separations in membranes [6].

∂u

∂t= D∆Ω(u3 − u− γ∆Ωu) (3.8)

(3.9)

where u is the concentration of the fluid, D the diffusion coefficient, andγ is related to the length of transition between fluid regions.

3.2 Patches and CAD

CAD is an acronym for Computer-Aided Design and a widely used toolby designers and engineers. It emerged in the 1960s from the US companyInternational Business Machines Corporation (IBM) which replaced andenhanced manual drafting. Today the technology has expanded to be avital part of the work flow for the development of new products. Thedesign process is eased since the drawn objects can be rotated to anyangle, become transparent, easily altered and reused. The drawn objects

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can be curves, surfaces and solids in both two-dimensional and three-dimensional space. In engineering, computer aided-design is often usedin conjunction together with Computer-aided engineering (CAE) whichincorporates tools like finite element analysis (FEA) or computationalfluid dynamics (CFD). An example of a 3D geometry in a CAD programis shown below in Figure 3.1.

Figure 3.1: Example surface created in a CAD program

In engineering a surface geometry is modelled in CAD as a partitioninto trimmed patches. The patches are defined with maps F from a refer-ence domain, which is non-curved. We define Ω to be a piecewise smoothsurface immersed in Rd, d ≥ 2. The geometry is assumed to have asmooth boundary with possible sharp edges, but no corners. We defineO = Ωi, i ∈ IΩ the set of subdomains Ωi called patches that makes upthe whole surface. Each patch boundary Γi = ∂Ωi is described by smoothcurves and the interfaces between patches in O is denoted as Γij . For each

patch Ωi we have a smooth injective map defined as F : Ωi → Ωi, whereΩi ⊂ I2 = [0, 1]2 is the reference domain. We illustrate a surface withpatches and interfaces in Figure 3.2.

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I2

Ω3

Ω1

Ω2

Ω5 Ω6Ω7

Ω4Ω3

Γ34F3

Figure 3.2: Representation of the surface via patchwise parametrizationswith a bijective map F .

3.3 Parametric Description

In our setting, each patch Ωi of the geometry Ω is a parametric surfacewhich is defined through its corresponding map Fi : Ω → Ωi. For eachpoint x ∈ Ωi we have the set of tangent vectors on the surface, i.e thetangent space defined as

Tx(Ωi) = span∂1Fi, ∂2Fi (3.10)

where ∂k is the partial derivative in the ek direction. The map F =(F 1, F 2, F 3) : Ωi → Ωi, can be written more explicit as

xyz

=

F 1(x1, x2)F 2(x1, x2)F 3(x1, x2)

(3.11)

where (x1, x2, x3) ∈ Ωi and (x1, x2) ∈ Ωi. Any tangent vector in Tx(Ωi)can be represented in terms of the local derivatives as

a = a1∂1F + a2∂2F =

∂1F

1 ∂2F1

∂1F2 ∂2F

2

∂1F3 ∂2F

3

[a1

a2

]= DF a (3.12)

since the derivatives spans the space.

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Metric. A metric is a function that measures distance between pair ofpoints in a set X. By using a specific metric, namely a metric tensor, wecan for instance obtain distances along curves through integration. Byequipping Tx(Ωi) with the usual Euclidean inner product we define theinduced inner product g on Tx(Ω) as

g(a, b) = a · b (3.13)

If we express a and b, above in (3.13), in the basis ∂1, ∂2 we can write

g(a, b) = a · b (3.14)

=

(DF a

)TDF b (3.15)

= aT Gb (3.16)

where G is the metric tensor with components gkl =(∂kF

)T (∂lF

). For a

given a ∈ Tx(Ωi) we can use the identities

a · b =(aT G

)b (3.17)

a · b = aT DF b =(DF Ta

)Tb (3.18)

to obtain the corresponding a as follows

a = G−1(DF Ti

)Ta (3.19)

The metric induces a norm as

‖v‖2g = g(v, v) (3.20)

with we can see that the mapping Tx(Ω) 3 a 7→ a ∈ Tx(Ω) is an isometry,i.e ‖a‖R2 = ‖a‖g and θ = θ.

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The gradient. The tangential gradient ∇u ∈ Tx(Ωi) can be expressedin terms of the basis ∂1Fi, ∂2Fi induced from the map F as

∇u = DFi∇u (3.21)

where ∇u is the local coordinates and can be obtain by taking the tangen-tial gradient of u in reference coordinates ∇ = ∂1e1 + ∂2e2 and applyingthe chain rule as

∇u = ∇(u Fi) = (DFi)T∇u = G∇u (3.22)

and thus the gradient coordinates in the reference basis is given by ∇u =G−1∇u

Integration. In the weak form (2.9) there are both surface integrals andline integrals. They are formulated over a possible curved surface Ω ⊂ R3

which could be complex to integrate over. This is resolved by utilizing themap structure F−1 : Ω → Ω and integrating over the reference domainΩ ⊂ R2 instead. The integral over ω ⊂ Ω can be written over ω ⊂ Ω as

∫

ωf(x) dx =

∫

ωf |G|1/2 dx (3.23)

where f = f F (x1, x2), dx = dx1dx2, and |G| = |det(G)|. Further letC be a curve in Ω parametrized by γ : [0, l]→ Ω then a line integral overthe curve C = F C is given by

∫

Cf(x)dC =

∫ l

0f γ(t)

∣∣ ddt

(F γ)(t)∣∣ dt (3.24)

=

∫ l

0f γ(t)‖γ′‖g(t)dt (3.25)

where γ′ = dγds is the unit tangent vector to the curve C.

The Method in Reference Coordinates. By using the parametricdescription we can reformulate the weak terms from (2.19) in reference

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coordinates and some examples are:

(f, w)Ω =

∫

Ωf w|G|1/2 dx (3.26)

(∇v,∇w)Ω =

∫

Ωg(G−1∇v, G−1∇w)|G|1/2 dx (3.27)

(n · ∇v, w)L2(∂Ω) =

∫

∂Ωg(n, G−1∇v)w‖γ′‖g dγ (3.28)

where g(u, v) is the metric (3.16) and

n =G−1ν

‖G−1ν‖g(3.29)

and ν is the unit normal to γ′ with respect to the Euclidean inner product.

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Chapter 4

Summary of Papers

Paper I. Cut Finite Element Methods for Elliptic Problems on Multi-patch Parametric Surfaces [12]: In this paper we develop a general frame-work for the Laplace-Beltrami operator on a patchwise parametric surface.On each patch there is a bijective map with a corresponding referencedomain. We use the cut finite element method together with Nitsche’smethod to enforce continuity over the interfaces between patches. A sta-bilization term is added to handle cut elements that might cause numericalerrors. Each patch map imposes a Riemannian metric, which we utilizeto compute quantities in the simpler reference domain. To be able tointegrate the terms accurately we have developed a quadrature formulawhich deals with cut elements. We show that the method is stable and wederive optimal order a priori estimates in the energy and L2 norms andalso present several numerical examples confirming our theoretical results.

Paper II. A Nitsche Method for Elliptic Problems on Composite Sur-faces [10]: We present a Nitsche method for diffusion problems on ge-ometries which consist of an arrangement of surfaces. In this settinginterfaces could have multiple (more than two) surfaces meeting. Theinterface could be any smooth curve, including triple points and sharpedges. This method avoids defining any co-normal to each interface, butinstead the corresponding co-normal to each surface is used in combinationwith the conservation law called Kirchhoff condition. Over each interfacethe sum of all co-normal fluxes should be zero. In this paper we show thatthis formulation is equivalent with the standard Nitsche interface method

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in flat geometries. We show different examples with matching meshes,non-matching meshes and cut meshes.

Paper III. Graded Parametric CutFEM and CutIGA for Elliptic Bound-ary Value Problems in Domains with Corners [13]: In this paper we havedeveloped a cut finite element method for elliptic problems with cornersingularities. We weakly enforce Dirichlet condition on the boundary andstabilize the cut elements using Ghost penalty. Our approach is to usean appropriate radial map that grades the finite element mesh towardsthe corner and this counter-acts the singularity in the solution. We provethat the method is stable by pulling back to reference domain and usingthe additional stability obtained from Ghost penalty. Optimal a priorierror estimates in L2 and energy norm are shown by utilizing a bound onderivatives of order k in the reference domain in terms of a weighted normin the physical domain which is bounded for the singular solution.

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Cut Finite Element Methods on Parametric Multipatch Surfaces

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