Eulerian-Lagrangian Localized AdjointMethod: The Theoretical Framework

Ismael HerreraInstituto de Geoffsica, UNAM, Apdo. Postal 22-582, 14000Mexico, Distrito Federal, Mexico..

Richard E. EwingCollege of Science, Texas A&M University, College Station, Texas 77843-3257

Michael A. CeliaWater Resources Program, Department of Civil Engineering andOperations Research, Princeton University, New Jersey 08544

Thomas F. RussellDepartment of Mathematics, University of Colorado atDenver, Denver, Colorado 80204-5300

Received 1 October 1992; approved without modification 15 October 1992

This is the second of a sequence of papers devoted to applying the localized adjoint method (LAM),in space-time, to problems of advective-diffusive transport. We refer to the resulting methodologyas the Eulerian-Lagrangian localized adjoint method (ELLAM). The ELLAM approach yields ageneral formulation that subsumes many specific methods based on combined Lagrangian andEulerian approaches, so-called characteristic methods (CM). In the first paper of this series theemphasis was placed in the numerical implementation and a careful treatment of implementation ofboundary conditions was presented for one-dimensional problems. The final ELLAM approximationwas shown to possess the conservation of mass property, unlike typical characteristic methods.The emphasis of the present paper is on the theoretical aspects of the method. The theory, basedon Herrera's algebraic theory of boundary value problems, is presented for advection-diffusionequations in both one-dimensional and multidimensional systems. This provides a generalizedELLAM formulation. The generality of the method is also demonstrated by a treatment of systemsof equations as well as a derivation of mixed methods. @ 1993 John Wiley & Sons, Inc.

~;

...INTRODUCTION

This is the second of a sequence of papers devoted to the application of the localizedadjoint method (LAM) to problems of advection-diffusive transport [1]. The LAM is anew and promising methodology for discretizing partial differential equations. It is basedon Herrera's algebraic theory of boundary value problems [2-6]. Applications have beenmade successively to ordinary differential equations, for which highly accurate algorithms

Numerical Methods for Partial Differential Equations, 9, 431-457 (1993)@ 1993 John Wiley & Sons, Inc. CCC 0749-159X/93/040431-27

432 HERRERA ET AL.

~~,.,.

were developed [5,7-9], multidimensional steady-state problems [10], and optimal spatialmethods for advection-diffusion equations [11-18]. Finally, this article and the companionone [1] (to be referred to as Paper I) provide generalizations of characteristic methods thatwe refer to as the Eulerian-Lagrangian localized adjoint method (ELLAM). Related workhas been published separately by some of the authors [19-22] and some more specificapplications have already been made [23-28].

The numerical solution of the advective-diffusive transport equation is a problem of greatimportance because many problems in science and engineering involve such mathematicalmodels. When the process is advection dominated the problem is especially difficult. InPaper I a brief review of the methods available was presented, from which we drawhere. The methods derive from two main approaches: optimal spatial methods (OSM) andcharacteristic methods (CM).

The first of these procedures employs an Eulerian approach and develops an accuratesolution of the spatial problem. For example, in the pioneering work of Allen and Southwell[29], a finite difference approximation was developed for the advection and diffusion termsthat gives exact nodal values for the simplified case of one-dimensional, steady-state,constant-coefficient advective-diffusive transport without sources, sinks, or reaction terms.More general and systematic results in this direction have been developed using the LAMapproach [5,7-10]. However, this kind of approximations tend to be ineffective in transientsimulations because of the strong influence of the time derivative. The salient features ofthis class of approximations may be summarized as follows: (i) Time truncation errordominates the solutions; (ii) Solutions are characterized by significant numerical diffusionand some phase errors; and (iii) The Courant number (Cu = V ~t/ ~x) is generallyrestricted to be less than 1, and sometimes much less than 1. A general comparisonof some of these methods was provided by Bouloutas and Celia [30].

Other Eulerian methods can be developed that perform significantly better than OSM~pproximations. These methods attempt to use the nonzero spatial truncation error(thereby differing from OSM) to cancel temporal errors and thereby reduce the overalltruncation error (see, for example, [31-33]). While improved accuracy results from theseformulations, they still suffer from strict Courant number limitations.

Finally, characteristic methods include many related approximation techniques that arecalled by a variety of names, such as Eulerian-Lagrangian methods (ELM) [34-37], trans-port diffusion method [38], characteristic Galerkin methods [39], method of characteristics(MOC) [40], modified method of characteristics (MMOC) [41-43], and operator splittingmethods [44-46]. Each of these methods has in common the feature that the advectivecomponent is treated by a characteristic tracking algorithm (a Langrangian frame ofreference) and the diffusive step is treated separately using a more standard (Eulerian)spatial approximation. These methods have the significant advantage that Courant numberrestrictions of purely Eulerian methods are alleviated because of the Langrangian nature ofthe advection step. Furthermore, because the spatial and temporal dimensions are coupledthrough the characteristic tracking, the influence of time truncation error present in OSMapproximations is greatly reduced.

This paper and its companion [1] provide a generalization of characteristic methodsusing the approach of the localized adjoint method. In Paper I, a specific space-time LAMformulation that naturally leads to generalized CM approximations (to be referred to asEulerian-Lagragian LAM:ELLAM) was introduced that is consistent and does not relyon any operator splitting or equation decomposition. In addition, all relevant boundaryterms arise naturally in the formulation; a systematic and complete treatment of boundaryconditions was presented which was shown to possess the global mass-conservation

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD 433

property. The numerical treatment and boundary conditions implementation was developedwith considerable detail.

The present Paper II dwells more on the general LAM theory, placing ELLAM morethoroughly in the general perspective of LAM. This contributes to a more complete pictureof the possibilities that should be explored and the problems that must be tackled inorder to make ELLAM a more effective modeling tool. The present paper begins byreviewing Herrera's algebraic theory as well as the LAM procedure. Then a generaldiscussion of the approach is presented, including illustrations of its application to ordinarydifferential equations. After this, the LAM formulation of the space-time of advection-diffusion equation is derived in one space dimension, and the choice of continuity andboundary conditions to be satisfied by test functions is briefly discussed. The ELLAMformulation is then extended to several space dimensions. To illustrate the generality ofLAM, the manner in which systems of equations are incorporated in this framework isexplained, and as an example, mixed methods are developed in this setting.

-i

II. GENERAL BACKGROUND

In this section, Herrera's algebraic theory of boundary value problems [2-6] is brieflyexplained.

Consider a region 0. and the linear spaces D} and Dz of trial and test functions definedin 0., respectively. Assume further that functions belonging to D} and Dz may have jumpdiscontinuities across some internal boundaries whose union will be denoted by I. Forexample, in applications of the theory to finite element methods, the set I could be theunion of all the interelement boundaries. In this setting the general boundary value problemto be considered is one with prescribed jumps across I. The differential equation is

;;£u = in in 0. , (2.1)

where 0. may be a purely spatial region or, more generally, a space-time region. Certainboundary and jump conditions are specified on the boundary ao. of n and on I,respectively. When 0. is a space-time region, such conditions generally include initialconditions. In the literature on mathematical modeling of macroscopic physical systems,there are a variety of examples of initial-boundary value problems with prescribed jumps.To mention just one, problems of elastic wave diffraction can be formulated as such[47,48]. The jump conditions that the sought solution must satisfy across I, in order todefine a well posed problem, depend on the specific application and on the differentialoperator considered. For example, for elliptic problems of second order, continuity of thesought solution and its normal derivative is usually required, but the problem in whichthe solution and its normal derivative jump across I in a prescribed manner is also wellposed [48].

The definition of a formal disjoint requires that a differential operator ;;£ and its formaladjoint ;;£* satisfy the condition that w;;£u -u;;£*w be a divergence, i.e.,

w;;£u -u;;£*w = V. {~(u,w)} (2.2)

for a suitable vector-valued bilinear function ~(u, w). Integration of Eq. (2.2) over 0. andapplication of the generalized divergence theorem [49] yields

( {w;;£u -u;;£*w}dx = ( ~a(w,u)dx + ( ~I(u,w)dx (2.3)Jn Jan JI

434 HERRERA ET AL.

where

~ii(U,W) = ~(U,W). n. and ~I(U,W) = -[~(u,w)]. n.. (2.4)

Here the square brackets stand for the "jumps" across I of the function contained inside,i.e., limit on the positive side minus limit on the negative side. Here, as in what follows,the positive side of I is chosen arbitrarily and then the unit normal vector n is takenpointing towards the positive side of I. Observe that generally !£u will not be defined onI, since u and its derivatives may be discontinuous. Thus, in this article, it is understoodthat integrals over n are carried out excluding I. Consequently, differential operators willalways be understood in an elementary sense and not in a distributional sense.

In the general theory of partial differential equations, Green's formulas are used exten-sively. For the construction of such formulas it is standard to introduce a decomposition ofthe bilinear function ~ii (see, for example, Lions and Magenes [50]). Indicating transposesof bilinears forms by means of an asterisk, the general form of such decomposition is

~ii(U,W) = ~(u,w). n. = ~(u,w) -Cfo*(u,w), (2.5)

where ~(u,w) and Cfo(w,u) = Cfo*(u,w) are two bilinear functions. When consideringinitial-boundary value problems, the definitions of these bilinear forms depend on thetype of boundary and initial conditions to be prescribed. A basic property required of~(u, w) is that for any u that satisfies the prescribed boundary and initial conditions,~(u, w) is a well-defined linear function of w, independent of the particular choice of u.This linear function will be denoted by gii [thus its value for any given function w will begii(W)], and the boundary conditions can be specified by requiring that ~(u,w) = gii(W)for every w E Dz (or more briefly: ~(u,,) = gii). For example, for the Dirichlet problemof the Laplace equation, it will be seen later that ~(u, w) can be taken to be uaw I an onan. Thus, if Uii is the prescribed value of u on an, one has ~(u,w) = uiiawlan for anyfunction u that satisfies the boundary conditions. Thus gii(W) = uiiawlan in this case.

The linear function Cfo*(u,.), on the other hand, cannot be evaluated in terms of theprescribed boundary values, but it also depends exclusively on certain boundary values ofu (the "complementary boundary values"). Generally, such boundary values can only beevaluated after the initial-boundary value problem has been solved. Taking the exampleof the Dirichlet problem for the Laplace equation, as before, Cfo*(u, w) = waul an and thecomplementary boundary values correspond to the normal derivative on an.

In a similar fashion, convenient formulations of boundary value problems with pre-scribed jumps requires constructing Green's formulas in discontinuous fields. This canbe done by introducing a general decomposition of the bilinear function ~I(U, w) whosedefinition is pointwise. The general theory includes the treatment of differential operatorswith discontinuous coefficients [5]. However, in this article, only continuous coefficientswill be considered. In this case, such decomposition is easy to obtain, and it stems fromthe algebraic identity:

.1

[~(u,w)] = ~([u],w) + ~(it,[w]),

where

[u] = u+ -u-, it = (u+u-)/2. (2.

The desired decomposition is obtained by combining the second of Eqs. (2.4) with (2.6)

~I(U,W) = ,:j(u,w) -~*(u,w), (2.:

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD... 435

with

..

:J(u, w) = -~([u], w) .Q (2.9a)

~*(u,w) = ~(w,u) = ~(u,[w]) °Q. (2.9b)

An important property of the bilinear function :J(u, w) is that, when the jump of u is

specified, it defines a unique linear function of w, which is independent of the particular

choice of u. When considering initial-boundary value problems with prescribed jumps, the

linear function defined by the prescribed jumps in this manner will be denoted by j~ [thus

its value for any given function w will be j~(w)] and the jump conditions at any point of Ican be specified by means of the equation :J(u, .) = j~. In problems with prescribed jumps,

the linear functional ~*(u, 0) plays a role similar to that of the complementary boundary

values '(6*(u, 0). It can only be evaluated after the initial-boundary value problem has been

solved and certain infomlation about the average of the solution and its derivatives on I

is known. Such information will be called the "generalized averages."

Introducing the notation

(Pu,w) = fnW:£Udx, (Q*u,w) = fnU:£*WdX, (2. lOa)

(Bu,w) = r S,8(u,w)dx, (C*u,w) = f '(6(w,u)dx (2. lOb)Jan an

(Ju,w) = f~:J(U,W)dX, and (K*u,w) = f~~(W,U)dX, (2.10c)

Eq. (2.3) can be written as

(Pu, w) -(Q*u, w) = (Bu, w) -(C*u, w) + (Ju, w) -(K*u, w). (2.11)

This is an identity between bilinear forms and can be written more briefly, after rearranging,

asP -B -J = Q* -C* -K*. (2.12)

This is the Green-Herrera formula for operators in discontinuous fields [2,6].The initial-boundary value problem with prescribed jumps can be formulated pointwise

by means of Eq. (2.1) together with

~(u, .) = gaand :J!(u, .) = jI. (2.13)

In order to associate a variational formulation with this problem, define the linearfunctionals f, g, j E D; by means of

(f, w) = { win dx, (g, w) = { ga (w) dx, (j, w) = {jI(w)dx. (2.14)In Jan JI

Then a variational formulation of the initial-boundary value problem with prescribed jumpsis

Pu = f, Bu = g, Ju = j. (2.15)

The bilinear functional J just constructed, as well as B, are boundary operators for P,which are fully disjoint. (For the definitions of the concepts that appear in italics here, thereader is referred to Herrera's original papers [2-6]). When this is the case, the systemof equations (2.15) is equivalent to the single variational equation

«P -B -J)u,w) = (f -g -j,w) Vw E D2. (2.16)

436 HERRERA ET AL.

This is said to be "the variational formulation in terms of the data of the problem," becausePu, Bu, and Ju are prescribed. Making use of formula (2.12), the variational formulation(2.16) is transformed into

«Q* -C* -K*)u,w) = (f -g -j,w) 'vw E D2. (2.17)

This is said to be "the variational formulation in terms of the sought information," becauseQ*u, C*u, and K*u are not prescribed. The variational formulations (2.16) and (2.17) areequivalent by virtue of the identity (2.12). The linear functionals Q*u, C*u, and K*usupply information about the sought solution at points in the interior of the region 0, thecomplementary boundary values at 00, and the generalized averages of the solution atI, respectively, as can be verified by inspection. of Eqs. (2.10), and as will be illustratedin the examples that follow.

Localized adjoint methods are based on the following observations. When the methodof weighted residuals is applied, an approximate solution u E Di satisfies

«P -B -J)u,wa) = (f -g -j,wa), a = 1,...,N, (2.18)

where {Wi,..., wN} C D2 is a given system of weighting functions. However, theseequations, when they are expressed in terms of the sought information, become

«Q* -c* -K*)u,wa) = (f -g -j,wa), a = 1,...,N. (2.19)

Since the exact solution satisfies (2.17) it must be that

«Q* -c* -K*)u, wa) = «Q* -C* -K*)u, wa), a = 1,..., N. (2~20)

Either in this form or in the form

«Q* -C* -K*)(u -u),wa) = 0, a = 1,...,N, (2.20')

Eqs. (2.20) can be used to analyze the information about the exact solution that is containedin an approximate one. In localized adjoint methods, these observations have been usedas a framework for selecting more convenient test functions.

III. IllUSTRATIVE RESULTS

As has already been mentioned, K*u supplies information about the average of thesolution and its derivatives across the surface of discontinuity I. Such inforn;ation can beclassified further. In particular, it is useful to decompose the averages K*u into"averagesof the function, the first derivative, etc. This is achieved by writing K* as the sum ofoperators KO*, K1*,..., each one containing the information about the average of thederivative of the corresponding order. Such decomposition is induced when ~*(u, w) isdecomposed pointwise into the sum of bilinear functions ~*(u, w), ~1*(U, w),. .., eachone containing the corresponding information pointwise. Similarly, 1 will be written as thesum of operators lo,P,..., each containing the jump of the derivative of correspondingorder, and j(u,w) will be the sum ofjO(U,W),jl(U,w), etc. When this is done,

K = LKi, 1 = LF, ~ = L~i, j = Lji. (3.1)i i i i

In view of Eqs. (2.20), it is clear that the information about the exact solution containedin an approximate one depends in an essential manner on the system of weighting functionschosen. The systematic classification of such information introduced by the algebraic

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD 437

theory can be used to develop weighting functions that concentrate the information ina desired manner. In very general terms, if one wishes to eliminate all the informationabout the sought solution in the interior of the region .0. -I (thus concentrating it inthe complementary boundary values and averages across I), the weighting functions mustbe chosen so that 0 = (Q*u, wa) = (Qwa, u). This condition is satisfied if Qwa = 0,i.e., ;£,*wa = O. Similarly, the information about the complementary boundary values canbe eliminated if the weighting functions satisfy cwa = 0 globally, i.e., C(g(wa,.) = 0pointwise. Elimination of the average of the function requires ~(wa,.) = 0; eliminationof the average of the first derivative requires that ~l(wa,.) = 0, etc. Note that boundarymethods are obtained when ;£,*wa = 0 and simultaneously '!l*(Wa,.) = O. Also, if allthe information is to be concentrated at one point, Green's functions must be constructed.However, development of such weighting functions requires solution of nonlocal problems.As a further general comment, it should be mentioned that, for partial differential equationsof second order with continuous coefficients, the condition ~l(wa,.) = 0 is equivalent to[w] = 0 (i.e., w is CO). This result, which will be shown later in this section for thespecial case of ordinary differential equations, implies that CO methods concentrate the in-formation on the values of the function, both at the interelement boundaries and in theinterior of the finite elements.

As an illustration of the procedure, let us review the results for general second-orderdifferential equations that are linear, which were derived by Herrera and co-workers[5,7-9] for one-dimensional equations. Actually, the methodology is applicable tomultidimensional equations of any order which may be time dependent, and to systemsof differential equations, as is explained in Sec. VII. However, for partial differentialequations (i.e., when more than one independent variable is involved), the implementationof the procedures is considerably more complicated and it is not possible to predict theexact values of the solution in general.

A physical situation that the general ordinary differential equation of second ordermimics is transport in the presence of advection, diffusion, and linear sources; a notationrelated with such processes will be adopted. The general equation to be considered is

;£,U = -d D~ -Vu' in n = [O,i]+ Ru = fn. (3.2a)dx \ dx

Attention will be restricted to the case when D and V are continuous (discontinuouscoefficients have been treated previously [5]). When the function u is assum~d to becontinuous, so that

(3.2b)

(3.2c)

reduces toau = 0 on I (3.2d)AX

by virtue of the assumed continuity of V and D.A partition {O = XO,Xl,... ,XE-l,XE = I} is introduced, which for simplicity is assumed

to be uniform, i.e., Xa -Xa-l = h is independent of a. Trial and test functions will beassumed to be sufficiently differentiable in the interior of each of the subintervals of thepartition, so that the differential operator is defined there and the jump discontinuities can

HERRERA ET AL.

-) -v ~ + Rw. (3.3)

only occur at internal nodes. This corresponds to taking I = {Xl,... ,XE-I} in the generalframework presented in Sec. II. Observe that the normal vector!:!. = 1 at 1, and !:!. = -1at O. On I the choice!:!. = I is convenient, because in this manner the positive side ofI is the side that is determined by the sense of the X axis. Suitable boundary conditionsare assumed to be satisfied at 0 and 1 in order to have a well defined boundary valueproblem. The boundary conditions can be Dirichlet, Neumann, or Robin [5], but they areleft unspecified, since the following developments accommodate any of them.

The formal adjoint of the operator ..<e, as defined by (3.2a), is

..<e*w = -!!-. (DdW __dw

dx dxTherefore

dU}-wD-

dxw;/'u -u;/,*w = d { ( dw -u D- + Vw

dx dx (3.4)

du(3.5)-wDd:;'

Application of Eqs. (2.9) yields

(3.6b)

( dw S;B(u, w) = u D"d:;- + Vw

(3.7)~,

Another possibility is

dw ( du )~(u W) = -uD- .n ~ (w U ) = -w D- -Vu .n., dx -" dx -

To be specific, only Eq. (3.7) will be used here. For Neuman problems, du/ dx is thedatum, so that

.

~(w,u) = ill D~ + Vw ~l(W,U) = -[W]D~,;

from which J and ~ are obtained by means of Eqs. (3.1). In Eqs. (3.6), an overbar is usedto indicate that the dot on top refers to the whole expression covered by the bar.

The definitions of the bilinear functions ~(u, w) and <Q,(w, u), depend on the type ofboundary conditions to be satisfied. For Dirichlet problems, u is prescribed and onepossibility is

In the most general form of Robin's boundary conditions, a linear combination of thederivative and the value of the solution are prescribed. This general case was developed in[5]. Here, attention is restricted to the case when the total flux Ddu/ dx -Vu is prescribed.Then

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD 439

nodes; by so doing, they were able to obtain algorithms that yield the exact values ofthe solution at those nodes. Using the developments already presented, derivation of suchalgorithms is a simple exercise. The information about the sought solution supplied by theweighted system of equations is exhibited in Eqs. (2.20). In particular, the information thatis supplied at internal nodes refers to the averages of the solution and its first derivativethere, given by ?J;'*u [see Eqs. (3.6b)]. Thus, in order to concentrate the information there,it is necessary to remove (Q*u, wa) and (C*u, wa) in Eqs. (2.20). This is achieved ifQwa = 0 and Cwa = O. The first of these conditions is equivalent to ..<£*w = 0, andthe other is the boundary condition «6(w,.) = 0, which in the case considered here mustbe satisfied at an = {O, I}. In view of Eqs. (3.7)-(3.9), the boundary conditions to besatisfied by the test functions are

dwD-:- + Vw dww=O

'dx dx

wherever Dirichlet, Neuman, and flux conditions, respectively, are prescribed on uObserve the "symmetry":

0, D 0, (3.10)

To eliminate the complementary boundary values from the weighted equations, the valueof the test functions must vanish wherever the value of the sought solution is prescribed.However, it is the "reserve flux" (Ddw / dx + Vw) which must vanish, wherever thefirst derivative of the sought solution is prescribed and conversely, it is the derivative(dw/dx) which must vanish wherever the flux (Ddu/dx -Vu) is prescribed.

Since our goal is to concentrate the information on the value of the sought solution atinternal nodes, taking into account that ~(w, u) = ~(w, u) + ~l(W, u), it is clear thatit is still necessary to remove the information about the first derivative. This will beachieved if the condition ~1 (w, .) = 0 is imposed on the test functions. In view of (3.6b)this condition is [w] = 0, on I.

In summary, the weighting functions that concentrate all the information in the valuesof the sought solution at internal nodes are solutions of the homogeneous boundary valueproblem with prescribed jumps:

;£*w = 0 on n,

[w] = 0 on I

on ao. = {O, l},~(w, .) = 0

(3.11)

-.,

Clearly, the condition [w] = 0 on L implies that wECO. Such weighting functions canbe taken having local support, because Caw/ax] * 0 is admissible [5,8]. Generally, thedimension of the space of solutions of Eqs. (3.11) is E -1, and if that space is usedto form the system of weighting functions in Eq. (2.18), an (E -1) (E -1) system ofequations possessing a unique solution is obtained for the average a of the approximatesolution at internal nodes. The uniqueness of solution of this system of equations impliesthat the averages predicted in this manner are equal to the averages of the exact solution atinternal nodes, since ~(u -u) = 0 in that case, by virtue of (2.20'). Also, at any internalnode, the value of the sought solution is equal to its average there, since it is continuous(Eqs. 3.2b), so that the exact value of the solution at internal nodes is predicted in thismanner.

A rigorous discussion of the conditions under which the resulting system of equationspossesses a unique solution requires the use of the concept of TH completeness. Thisconcept was introduced by Herrera in [2,51], where he presented a rigorous discussion of

440 HERRERA ET AL.

this question in an abstract setting allowing considerable generality, since the conclusionsthat he obtained are independent of the order of the differential equations and the numberof independent variables involved. However, attention was restricted to the case whenthe differential operator is symmetric and the corresponding analysis for nonsymmetricoperators is wanting. This matter is being studied at present and will be addressedelsewhere.

A similar procedure can be developed for obtaining the exact values of the first derivativeat internal nodes, the main difference being that one must require that ~(w, .) = 0 insteadof ~l(W,.) = O. In view of Eq. (3.6b), Eq. (3.11) is replaced by

;;£*w = 0 on n,

dw-

on an = {O.l}.C(6(w,.) =0

on I-]=0[Ddw/ (3.12)dxThe corresponding algorithm was developed in [5,8]. Combinations in which the valueof the solution is obtained at some nodes and its derivative at others, or algorithms thatsimultaneously yield the exact values of the solution and its derivative at internal nodes(with a correspondingly larger system of equations to be solved), are also possible [5].

Until now, no specific representation of the approximate solution has been adopted.Let {<I>°, <1>1,.. ., <pN} be a system of trial functions, generally fully discontinuous (i.e.,the function and its derivative have jump discontinuities at internal nodes), and letU = LAj<l>j be an approximate solution satisfying Eqs. (2.18) or, equivalently, (2.19).When the system of weighting (or test) functions fulfilling (3.11) is TH complete, then

a(Xj} U(Xj),,E - (3.13a)j=

where u(x) is the exact solution. Correspondingly, if the system of weighting functionssatisfies (3.12) and is rn complete, then

It is important to observe that either Eq. (3.13a) or (3.13b) holds independently of the trialfunctions used. In particular, they can be fully discontinuous and they can also violate theprescribed boundary conditions, although this would produce poor approximations, exceptat the nodes. This is discussed further below.

It is worth exploring some of the implications of these results. To relate the resultsthus far obtained to standard variational formulations used in finite element meth-ods, let us consider the one-dimension (ill) version of the Poisson equation [i.e.,Eq. (3.2a), with D = 1, V = R = 0], subject to homogeneous Dirichlet boundary condi-tions [u(O) = u(l) = 0]. A standard variational formulation for this problem is as follows:u E HJ([O, I]) is a weak solution if and only ifi 1 du dw 11

---dx = Jow dx V' CI) E HJ([O, 1]), (3.14)0 dx dx 0

where HJ([o, I]) is the subspace of the Sobolev space HI ([0, I]), whose members havevanishing traces.

Let us look for an approximate solution it using trial functions {cI>l,..., cl>E-l} which arelocally linear and globally continuous [Fig.1 (a)]. In addition, use the same collection as testfunctions, as is usually done when applying Galerkin method. Let it(x) = IJ::t U jcl>j(x).

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD... 441

~o 'L:::~::::~ :'0 Xi-I Xi XI+1 L

(a)

4>+=-1

(b)

FIG. 1. (a) Base function: Locally linear, globally continuous. (b) Base function: Locally linear,discontinuous.

Then <l>j, for j = 1,..., E -1, as well as 12, belong to HJ([Q, 1]). Observe that for thisclass of functions one has

= «P -B -J)u,w) (3.15)

Here, Eqs. (3.6a) and the facts that u and x are continuous and vanish on the boundarywere used. Equation (3.15) illustrates the fact that the standard variational formulationfor the Laplace operator is a particular case of the general variational principle (2.16) interms of the data of the problem. However, the standard variational formulation can onlybe applied when both trial and test functions are continuous, while the algebraic theorysupplies a systematic manner of extending it to cases where trial and test functions arefully discontinuous.

For the case where the prescribed boundary conditions are nonhomogeneous, a suitablerepresentation of the approximate solution is

= rXa+l}Xa-l

Ua+l + Ua-l -2Uafnwadx, a = 2, ,E -2, (3. 17a)h

Let us apply the variational formulation (2.18) using the weighting functions{ cI> 1, ..., cI> E -1 } associated with internal nodes. This system concentrates all the information

at internal nodes because it satisfies Eqs. (3.11); moreover, it is 111 complete. The resultingsystem of equations is

442 HERRERA ET AL.

fulfilled by the exact solution, one has

Ua+l + Ua-l -2Ua Ua+l + Ua-l -2ua a = 2,...,E -2 (3. 18a)=h

together with

UE-2 -2UE-l~ UE-2 -2UE-l U2 -2U1 U2 -2Ulh h h (3.18b)

which implies Ua = u(xa) for a = 1,...,E -1 (i.e., at all the interior nodes), aspredicted by the theory.

The above properties depend solely on the weighting functions and are independent ofthe trial functions used. In particular, when the boundary values are not satisfied by thetrial functions [i.e., Uo * UO, UE * U/ in Eq. (3.16)], one would expect to obtain an(E + 1) (E -1) system of equations that would be undetermined, since the number ofunknowns is greater than the number of equations. However, Uo and U E do not occur in thesystem that is obtained, and the resulting system can be interpreted as an (E -1) (E -1)system for {Ut,..., UE-t}, whose only solution, as already mentioned, is the values of theexact solution at the internal nodes, leaving Uo and UE undetermined. This latter fact isnatural, since the system of equations (3.18) was derived using the variational principle(2.17), in terms of the sought information, and the values of the sought solution at theboundary are not included in the sought information. Indeed, in the boundary only thederivative is included in the sought information.

Moreover, the trial functions themselves can be changed arbitrarily and still thenodal values will be predicted correctly. In particular, let us illustrate the use of fullydiscontinuous trial functions by exhibiting these results when such trial functions are used.To this end, keeping the same weighting functions as before, change the trial functions inthe representation (3.16) of the approximate solution, to [see Fig. 1(b)]

.-3(x -Xj-t)

=

h

Xj-l S x S Xjh

X -Xj+l<l»j(x) = (3.19)Xj :5 x :5 Xj+lh

0, elsewhere.

Then the same system of equations (3.17) is obtained. Observe ti!at the averages of thetrial functions given by (3.19) are zero at internal nodes, except <f>j(Xj) which is equal toone. Thus (3.17) is equivalent to

....ua+! + Ua-l -2ua -~= rxa+1

}Xa-liowa dx, a = 2, ,E -2, (3.20a)h..;4E-2 -2;4E-1

..U2 -2Ul

h

l X2 fnWldx -~

0 h'-

h

f XE

XE-2

whose only solution is a(xa) = u(xa) for a = 1,...,E -1 [i.e., Eq. (3.13a)]. Thus, evenif discontinuous trial functions are used, the values of the exact solution are predictedcorrectly by the averages at internal nodes of the discontinuous approximate solutions.

Although very simple examples were chosen to illustrate the results presented in thissection, the conclusions are valid for the general Eq. (3.2a) and also for the different kindsof algorithms that were introduced. They exhibit the general fact that the information about

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD... 443

~,.

the sought solution contained in an approximate one is independent of the trial functionsused. However, the reader must not be misled to conclude that the choice of trial functionsis irrelevant. On the contrary, as is well known, a judicious choice of the trial functions isessential to develop satisfactory approximate solutions. For example, if the trial functions(3.19) are used as interpolators in present examples, very poor estimations of the solutionswould be obtained in the interior of the subintervals [Xi-I, Xi], in spite of the fact that itsvalues at internal nodes are predicted with unlimited precision.

However, an important conclusion that can be drawn from the preceding examples is thatin the construction of approximate solutions, there are two processes, equally importantbut different, that must be clearly distinguished. They are (i) gathering information aboutthe sought solution, and (ii) interpolating the information about the sought solution whichis available.

These two processes are distinct, although in many numerical methods they are notdifferentiated clearly. The information that is gathered is determined by the weightingfunctions, while the manner in which it is interpolated depends of the trial functionschosen. A peculiarity of the examples that have been given in this section is that here,those processes are not only independent, but they need not be carried out simultaneously.Indeed, since the exact values at the nodes are obtained independently of the trial functionsused, given that the requirements in Eq. (3.11) are satisfied, one can obtain them first andchoose the interpolator afterwards.

Also, the two processes mentioned above are to a large extent independent, exhibitingsome of the severe limitations associated with methods such as the Galerkin method,in which trial and test functions are required to be the same. The conditions that testfunctions must satisfy in order to be effective for gathering information, in general, will bequite different from those that must be satisfied by trial functions in order to be effectiveinterpolators.

A point that deserves further attention is the criteria that must be used to judiciouslyselect effective trial functions. Taking into account their role as interpolators, it is clear thatapproximation theory must be applied. However, in many cases the matters involved maygo beyond approximation theory. For example, in the illustrations presented thus far inthis section, which in some sense are. extreme cases since the exact values of the solutionare obtained at the nodes, a very efficient way of interpolating the available informationwould be to solve the boundary value problem that is defined by that information on eachof the subintervals [Xi-I, Xi]. These questions, although important, are complex, and itwould not be appropriate to explore them in all their generality at this point. Thus weleave the matter here, but intend to resume it elsewhere.

IV. ADVECTION-DIFFUSION EQUATION

When applying the methods of Sec. II to time-dependent problems, it will be necessaryto consider a region n in space-time. Also, the surface I on which discontinuities canoccur will be a surface in space-time, and a suitable notation will be required. Space-timevectors M will be written as pairs:

,..M = ~,mt), (4.1)

where!.!! is the vector made by its spatial components and mt corresponds to its temporalcomponent. Let ~I be the vectorial velocity of the discontinuity I(t), where I(t) is theset of points of I whose time coordinate is t. This is a vector in space, which can be

444 HERRERA ET AL.

written as

YI = VIQ, (4.2)

where!.! is the unit normal vector to I(t). Generally, VI can be positive or negative,depending on the sense of motion of I(t) and the choice of n. In particular, in the one-dimensional case, !.! will be taken to be equal to 1 on I. Observe that the space-time vector(yI' 1) = (VIQ, 1) is tangent to I. Using this fact, it is easy to see that a space-time unitnormal vector ~ to I is given by

( 2) -1/2 N = 1 + VI ([!, -VI), (4.3)

,,.au

ax-VU) + Ru = fo(x,t) in.o.,

x E fix = [0,1],

t E fit = [tn, tn+l]

(x, t) E fi = nxXnt,

subject to initial conditions.#

u(x,tn) = un (X) (4.5)

and suitable boundary conditions at x = 0 and t. The following development accommo-dates any combination of boundary conditions. The manner in which the region nandthe initial conditions were chosen in Eqs. (4.4) and (4.5) is convenient when applying anumerical integration procedure step by step in time.

The adjoint operator is

.aw a( aw ) aw ~ w := ---h va; -va; + Rw

at

and ~(u, w) as defined by Eq. (2.2) is

~(u,w) = {U(D~~ + vw ) -WD~,uw }ax ax

-wD~+ (V -VI)Wax

(4.8)Assuming that the physical process which Eq. (4.4) mimics is that of transport with Fickiandiffusion of a solute whose concentration is u in a free fluid moving with velocity V, thesmoothness conditions implied by mass balance are [48]

auu(V -VI) -D- =0 onI

axIn addition, Fickian diffusion implies

[u] = 0 on I. (4.10a)

In this section, we consider the one-dimensional transient advection-diffusion equationin conservative form:

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD... 445

When the coefficients V and D are continuous, Eq. (4.9) in the presence of (4.10a) canbe replaced by

ouox

=0 on! (4. lOb)

Using Eqs. (2.9), it is seen that

(4.11a)

(4.11b)

, (4. 12a)

on ann, (4.13a)~(u,w) = -uw

on On+1O, (4.13b)CQ,(w, u) = -uw

iJwSE(u,w) = uD ax (4. 13c)~,

on aNn. , (4. 13d)t!

It will be useful to decompose the boundary ao. into aoo., a/o., ano., and an+lo.,which are defined as the subsets of 0. for which (x, t) satisfies x = 0, x = 1, t = tn,and t = tn+l, respectively. The initial conditions given by Eq. (4.5) are to be satisfiedat ano., and the boundary conditions pertain to aoo. u a/o.. These latter conditions canbe of Dirichlet (u = uo), Neuman [D(aujax)fl. = q], or Robin type, or a combinationof them. Because of the special role that the total flux Dauj ax -Vu plays in massconservation, the only boundary conditions of Robin type that will be considered will bethose for which (Daujax -Vu)fl. (= F) is prescribed. In what follows, the notationsavo., aNo., and aFo. refer to that part of the boundary where Dirichlet, Neuman, andtotal-flux boundary conditions are prescribed, respectively.

The bilinear functions S:B(u,w) and ~(w,u) implied by the initial and boundaryconditions are

446 HERRERA ET AL.

natural for an initial value problem. Also, in the case of Dirichlet boundary conditions,alternative expressions to (4.13c) are

~(u.w) = U(D~ + VW)n.. (4.13f)

However, in this paper, we use (4.13c) only. In view of (4.11) and (4.13), it is clear that

giJ(W) = -unW on ann, on aDD. , (4.14a)ga(W) = Ua(D~ + Vw) .nax -

on aNn, giJ(W) = -wF .~ on oFfi, (4. 14b)ga(W) = -wq .!!.

,while jI(W) = 0 on I. The expressions for the bilinear functionals B, C, J, and K, areobtained by integration of~, «6,:1, and~, respectively, the first two on aO, and the lattertwo on I. Similarly, according to (2.17), the expressions for f, g, and j are obtained byintegrating In,go, and jI, in 0" ao', and I, respectively. In the present case jI = 0,so that j = 0 also.

As in Sec. III, a partition of [0, l] is introduced and the region 0, is decomposedinto a collection of subregions 0,1,..., o'E (Fig. 2), limited by space-time curves Ii,i = 1,..., E, whose parametric representations are given by the functions O"i(t); it will beassumed that discontinuities occur exclusively on these lines i.e.,

E

I=UIi.i=1

In addition, it is assumed that each such curve passes through its corresponding node attime tn+1 [i.e., O"i(tn+1) = Xi] and the notation x; = O"i(tn) is adopted. The velocity ofpropagation VI of each of these lines of discontinuity is dO"i/dt.

Using Eqs. (4.11) and (4.12), the expressions for J and K* can be obtained by integration;they are the sum of the contributions of each of the curves Ii. Thus, one can write

E] = L fa

a=!K*and (4.15)

FIG. 2.

XI-3 X 1-2 X 1-1 XI XI+l

Space-time supports of weighting functions.

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD.. 447

where

(Jau,w) = JIa![UJD~ -W(D[~ -(v -VI)[UJ)!adt, (4. 16a)

(K;u,w) = JIa 1"[ D~x-] -[wJ(D~ -(V -VI)" )!adt. Here the subindex Ia means that the line integral is to be carried out on Ia. To obtain

Eqs. (4.16), use has been made of the fact that on each line Ia, the element of time dt is

(1 + Vi)-l/2 times the length of the element in space-time.

In a similar fashion, it is convenient to decompose the bilinear functions B and C* into

the contributions which stem from ann, an+ln, aDn, aNn, and aFn. In this manner one

can write

B = Bn + BD + BN + BF

aw(4.16b)

and

C*=C:+1+C;'+C;;'+C;, (4.17)

f where

(Bnu, w) = -11 (UW)t=tn dx, (4.18a)

f aw (BDu,w) = uD- .n dt,

aDo ax-

(C~+lU,W) = -11 (UW)t=tn+ldx,

(C;u,w) = (- VU) .~dt,Jaoo

~

( au wD-

ax

(4. 18b)

(BNu, w) = -iNn (C~u,w) = -( U(D~J aNn ax

auwD- .ndtax -,

(BFu, w) = -( W(D~ -VU ) .ll.dt,J OFO ax

aw

ax(c;u, w) = - !!dtuD

iFfi

wq .n.dt, wF' Qdt (4.19b)(gN, W) = -iNn

V. EULERIAN-LAGRANGIAN LAM

In what follows, the variational formulation in terms of the sought information, Eq. (2.17),

«(Q* -C* -K*)u,w) = (f -g -j,w), (5.1

will be applied. In addition, weighting functions w will be chosen satisfying Qw = 0, i.e.,

CD* aw a ( aw ) aw .n.;L W = D- -V- + Rw = 0 10 ~£. (5.2

at ax ax ax

In this case (5.1) becomes «(C* + K*)u, w) = (g + j -f, w).

(4. 18d)To complete the formulation of the problem, it remains to define the linear functionals

t, g, and j. The last one is zero, while g = gn + gD + gN + gF, with

448 HERRERA ET AL.

Two important differences between the present case and the simple developments forordinary differential equations that were presented in Sec. III deserve attention. In thecase of ordinary differential equations, obtaining information about the sought solution (orpossibly its derivative) at internal nodes was our main goal. Thus algorithms for whichall the information was concentrated at internal nodes were developed, and by doing so,it was possible to predict the exact values of the sought solution at such nodes. This wasfeasible because TH-complete systems for ill problems are finite dimensional. As opposedto such simple developments, TH-complete systems for a partial differential equation suchas (4.4) are infinite dimensional. However, only a finite number of test functions can beused. Different choices of test functions that satisfy Eq. (5.2) lead to different classes ofapproximations, including optimal spatial methods and general characteristic methods.

On the other hand, when a numerical integration procedure is applied to Eq. (4.4), stepby step in time, the objective is to predict the values of u at time tn+l, when the valuesat time tn and the boundary conditions are given. Ideally, all the information about thesought solution should be concentrated in the value of the solution u at each one of thesubintervals [Xi-I, Xi], i = 1,... ,E, at time tn+l. For example, our goal could be obtainingthe ;£2([0, /) projection of the exact solution u(x, tn+l) on the subspace of piecewise linearinterpolators that are globally continuous. This subspace of ;£2 ([0, /) is generated by thesystem of functions

t

x -Xj-l

I1.xXj+l -X

Xi-! S X S Xi

Wi(XI. tn+l) (5.3)l Ax x ~ x ~ Xi+l .

This requires elimination of all information about the sought solution except at suchsubinterval and time. The weighting functions that do such job, in addition to satisfying;£*Wi = 0 in 0., must be smooth (i.e., [Wi] = [awi / ax] = 0) in the interior (0, f,) X(tn, tn+l) of 0. and must fulfill the boundary conditions Cfo(w,.) = 0 on the lateralboundary of 0., where Cfo(w,,) is given by Eqs. (4.13c)-(4.13e). Also, Wi (x, tn+l) = 0,except when x E [Xi-I,Xi+I]. In the interval [Xi-I,Xi+I], one requires that Wi (X, tn+l)be given by Eq. (5.3), by virtue of (4.13a). Then the resulting initial boundary valueproblem generally will be well posed [49], but such a weighting function would benonlocal.

Generally, in numerical applications, localized weighting functions are sought. At ageneral interior node Xi, as the one illustrated in Fig. 2, such localization can be achievedby introducing nonsmooth weighting functions. Thus, if the condition ;£*Wi = 0 issustained, then either [awl/ax] :#: 0, or [w] :#: 0, or both, and some information aboutthe solution u, or its normal derivative, or both, on the curves Ij, where discontinuitiesoccur, will be incorporated in the final system of equations. This is so in spite of thefact that the actual objective is to obtain information about the sought solution at timetn+l. The classification of numerical methods into OSM and CM can be related to thespeed of propagation of discontinuity lines. If time-independent solutions of Eq. (5.2)are chosen as weighting functions, then VI = 0 necessarily, and one is led to optimalspatial methods, to which several papers have been devoted using the LAM approach[12-18]. On the other hand, if the lines Ij satisfy VI = V, characteristic methods areobtained.

As mentioned above, there are also several possibilities for the degree of smoothnessof the weighting functions. In Paper I, weighting functions satisfying the condition[w] = 0 were chosen. In view of Eq. (4.12b), it is clear that for this special choice,9J:'1(W,u) vanishes identically, and that, in the lines of discontinuity Ii (i = 1,...,E),

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD... 449

all the information is concentrated in the sought solution u. In this case Eq. (4.12a)becomes

-ax

Assume that our goal is, as before, to obtain the ;;£2([0, /]) projection of the exactsolution u(x, tn+l), on the subspace of piecewise linear interpolators that are globallycontinuous. In the spirit of the previous developments and taking into account the limitation[ow/ow] * 0, which is unavoidable, a suitable set of properties for the test functionsWi (x, f), is the following:

(a) Support of Wi is Oi = of u O~, where of = 0; and O~ = 0;+1 (i = 1,...,E -1). See Fig. 2.

(b) Wi satisfies Eq. (5.2), i.e., .<£ *Wi = 0 in O.(c) At t = tn+1, Wi reduces to the piecewise linear interpolator given by Eq. (5.3).(d) Wi is continuous.(e) The jump [iJwijiJX] is constant on Ii.(f) At the lateral boundary of 0, boundary conditions which eliminate all the boundary

information [i.e., C(6(Wi, .) = 0] are imposed.

By inspection of Eqs. (4.13c)-(4.13e), it is seen that the last condition is

Wi = 0 on aDo. , (5.5a)awi -

D- + Vw' = 0 on aNn

ax- ,OWi = 0 on of.o. .'~.~-'ax

Observe that in the case of Dirichlet boundary conditions, Eq. (5.5a) is to be applied evenif the option (4.13f) for defining ~ and C(6 is used.

The development of test functions with these properties is not easy, in general, when thecoefficients are nonconstant, even if the domain .0. i does not intersect the lateral boundaries,

but may become particularly involved when the domain intersects one of the lateralboundaries. For the case when the coefficients of Eq. (4.4) are constant, the source termvanishes (R = 0), and the partition is uniform, the test functions used in Paper I, were

(x, t) E o.~

Wi(X, t) =

(x,t)En~

If the domain .0. i does not intersect the lateral boundaries, these weighting functions satisfy

all the required properties, (a)-(f) above; however, if the lateral boundary is intersectedby the corresponding domain, then (f) is violated.

An important advantage of the ELLAM approach is precisely its ability to deal withboundary conditions effectively. As was demonstrated through numerical examples inPaper I, the ELLAM approach provides a systematic and consistent methodology forthe proper incorporation of boundary conditions. This allows construction of an overall

450 HERRERA ET AL.

The jumps are

X 1-1 X.I Xi+\tn+1

Ij-1 Ii Ii+l

t*

t"Xc X*j X*i+l

FIG. 3. Case when the support of Wi intersects the inflow boundary

approximation that possesses the conservative property, thereby assuring conservation ofmass in the numerical solution.

Observe that for such weighting functions, ,11 and ~ do not vanish on three lines ofdiscontinuity, at most Ii-I, Ii, and Ii+l. Thus

This equation follows from Eq. (15b) of Paper I. The additional terms relative to Eq. (5.9)are due to nonzero Wi at x = 0 [i.e., they are due to violation of property (f) above].However, this leads naturally to the presence of the total flux term (avu -Dau/x) atthe inflow boundary. This is physically appropriate, leading to global mass conservation,

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD.. . 451

and the resulting set of equations yields accurate numerical results [1]. An alternativeformulation with very similar properties, based on integrating by parts once instead oftwice, is developed in [19], also with accurate numerical results; error estimates of optimalorder based on this formulation are proved in [52]. As mentioned in Paper I, the integralsthat appear in these equations may be approximated in many different ways. Differentapproximations of these integrals lead to different CM algorithms reported in the literature.

VI. MULTIDIMENSIONAL ADVECTION-DIFFUSION EQUATION

Application of the algebraic theory to multidimensional problems is straightforward.Only slight modifications have to be made in Eqs. (4.4)-(4.12). The region Ox ismultidimensional in this case, and the equations are

:£u = au + V (Vu) -Ru -V (D. Vu) = fn(x, t) -" ~,in 11at '-' J'"

The initial conditions (4.5) are sustained. The adjoint operator is

CD* aw -~ --.;Lw=- -v .Vu -Rw -V CD Vw)at -

~(u, w), as defined by Eq. (2.1), is

~(u,w) = {uDVw + Vw) -wDVu, uw}

Therefore

dW

~I(U,W) -[<2i>(u,w)] N= U(D + (v -V~)w' -WD~ ] , (6.4)an an

with the same assumptions as before. The smoothness conditions implied by mass balanceand Fickian diffusion are

au

an[u] = = 0 on I

Using Eqs. (2.8), it is seen that

+ w(V -VI)!. (6.6a)

(6.6b)

+ (V -VI)[wJ },.L on

~l(W,U) = -(I + Vi)-lf2[wJD~. (6.Th)on

Finally, the bilinear functionals ~(u, w), and Cfo(w, u), associated with initial and boundaryconditions, are

!i~,~(u,w) = -, -~"",

(6.7a)

on ann, on On+l!1, (6.Ma)-uw

~(u,w) = uD~ (6.8b)an'

C(6(w, u) = -uw

HERRERA ET AL.

VII. SYSTEMS OF EQUATIONS: MIXED METHODS

As mentioned above, localized adjoint methods are very general. In particular, they areapplicable to systems of equations. This section is devoted to explaining such applicationsand to developing mixed methods as an illustration.

A. Systems of Equations

Let u be a vector valued function defined in the region n. A linear system of equationscan be written as

in n,~u = fo (7.1)where ~ is a linear differential transformation. The adjoint ~* of ~ is defined by thecondition

{~(u.w)}.~*w = v (7.2)w ~u -u

where ~(u, w) is a vector-valued bilinear function.Application of generalized divergence theorem yields

( {w .~u -u .~"w}dx = ( 9R.o(u,w)dx + ( 9R.I(u,w)dx,Jn Jon JI (7.3)

~iJ(U,w) = ~(U,w) .Q and ~(u,w) = -[~(u,w)] .Q.

These equations are very similar to Eqs. (2.2) and (2.3), and the other equations of Sec. II

[(2.4)-(2.8)] are essentially the same. Thus, associated with every kind of boundary

conditions, one has a decomposition

~iJ(U,w) = ~(u,w)' Q = ~(u,w) -C{6(w,u), i: where ~(u,w) and C{6(w,u) are two bilinear functions. Using the identity

[~(u,w)] = ~([u],w) + ~(ti,[w]),

(7.4)

(7.5)

(7.6)

it is seen that

~I(U,W) = :f'(U,W) -~(W,U), (7.7)

1(u,w) = -~([u],w) .!!,

~(w,u) = ~(u,[w]) .!!.(7.8a)

(7.8b)

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD

B. Mixed Methods

As an illustration of how LAM can be used to provide insight into mixed methods, considerthe steady state of the general advection-diffusion equation of Sec. IV,

V .(DVu) -V .(Yu) + Ru = fo(x) in 11. (7.9)

This can be replaced by the system of equations

I!.. -D1/2Vu + D-1/2yu = 0, (7. lOa)

V. (DI/2I!..) + Ru = fo(:!.). (7. lOb)

In three spatial dimensions, Eqs. (7.10) constitute a system of four e~uations (underlinedquantities are 3 D vectors). Thus, a four-dimensional vector u = {I!.., uJ will be considered,

and the linear differential operator ;£,

~u = {I!.. -D1/2Vu + D-1/2yu, V .(DI/2I!..) + Ru}. (7.11)

The identity

(7.13)w .~u -u

implies that

.@.(U,w) = WDl/2E. -uDl/2~. (7.14)

The smoothness conditions implied by conservation of mass are

[u] = 0 and [Dl/2E.]. fl. = O. (7.15)

When the coefficients are continuous, the latter of these equations is equivalent to[E.] .fl. = O. Use of Eqs. (7.8) and (7.14) leads to

(7.16a)°ll.,

I .n.. (7.16b)

Equation (7.16b) has interesting implications. Because the exact solution u = {E., u}

satisfies (7.15), one has p .n. = p .n. and it = u. If it is desired to concentrate all

the information in the flux at interelement boundaries, then the weighting functions

w = {1, w}, in addition to satisfy the complementary boundary conditions «6(w,.) = 0,

must satisfy the adjoint system of differential equations ~*w = 0, which is

1- D1/2Vw = 0, (7. 17a)

V .(Dl/21) + D1/2~. 1 + Rw = O. (7.17b)

By inspection of Eq. (7.16b), it is seen that elimination of the information about thefunction u requires that [q] .n. = 0, i.e., that the normal component of q be continuous.

However, the weighting function w must be discontinuous. This is essentially the Raviart-

Let w be a function whose values are the four-dimensional vectors {1, w }. Then theadjoint differential operator is defined by

454 HERRERA ET AL.

Thomas [52] result, which constitutes the basis of mixed methods. This result has beenquite useful for approximating the total flux directly as an independent variable. Inparticular when D = l,:r = 0, and R = fn = 0, Eq. (7.9) becomes the Laplace equationwhile (7.17) can be written as

~=Vw, v 1=0 (7.18)

Thus q must be incompressible with continuous normal component across interelementboundaries, while w is discontinuous.

VIII. DISCUSSION AND CONCLUSIONS

In a sequence of two papers, localized adjoint method has been applied in space-time to problems of advective-diffusive transport. The approach is based on space-timediscretizations in which specialized test functions are applied. These functions satisfythe homogenous adjoint equation locally within each element. The resulting method isreferred to as the Eulerian-Lagragian localized adjoint method. The ELLAM approach, inaddition to providing a unification of characteristic methods (CM), supplies a systematicframework for incorporation of boundary conditions in CM approximations. Any type ofboundary conditions can be accommodated in a mass conservative manner. This seemsto be the first complete treatment of boundary conditions in Eulerian-Lagragian methodsthat leads to a conservative scheme for the general transport equations. Additionally, theELLAM approach provides a framework within which LAM concepts can be applied toadvection-dominated problems, handling time-dependent situations more accurately thanOSM. Thus ELLAM combines Eulerian-Lagrangian ideas and the LAM framework totheir mutual benefit.

In Paper I [1], the numerical implementation was developed and discussed thoroughly. Inthis second paper of the series, the theoretical aspects are covered in a more complete form,and the ELLAM procedures are more clearly related with the general LAM framework.This provides a more systematic development of the ELLAM methodology, making itpossible to establish a more complete picture of the possibilities that should be exploredand the problems that must be tackled in order to make ELLAM a more effective modelingtool. In particular, the LAM framework has been demonstrated to be very suitable formotivating specialized test functions. The effect that different boundary and continuity(or smoothness) conditions, satisfied by test functions, have on approximate solutions isclearly exhibited. Also, the LAM framework leads in a natural manner to a definitionof suitable unknowns for a given problem. For example, when developing the numericalimplementation of ELLAM in Paper I, it became apparent that in some cases it wasnecessary to introduce the total flux as an additional unknown at the boundaries, in spiteof the fact that the main goal was to predict the value of the function at time tn+l.The generality of the theory was corroborated, once more, by applying it to systems ofequations and deriving mixed methods.

However, there are many points that should be studied in more depth. We need a moreextensive study of both the theory and implementation of ELLAM techniques for variablecoefficients, particularly in multidimensional applications. Implementation of boundaryconditions for variable-coefficient problems in multiple dimensions is also an importantproblem. In addition, the treatment of nonlinear problems deserves further study. Sincethe unknown variables appear in nonlinear coefficients that are usually evaluated in theinterior of mesh blocks via numerical quadrature, greater attention must be placed on

EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD 455

the full approximation-theoric properties of the trial functions in these applications. Thepotential of local refinement in both space and time holds enormous potential for ELLAMand is the object of ongoing research.

Finally, we want to emphasize that ELLAM forms a general and powerful framework forinvestigating and comparing a wide variety of numerical methods for problems that haveimportant advective properties. The framework motivates different choices of test functionsto approximate different properties of the unknowns or even different unknowns, such asfluxes. The general theory is expanding to provide more insight. These techniques appearto have enormous flexibility and potential for treating general advection-diffusion-reactionproblems.

This work was partially supported by the International Atomic Energy Agency underContract No. 6088jRB, by the International Development Research Centre, Canada, underContract Nos. 89-1029-02, and by the National Science Foundation under ContractsNos. DMS8712021, RII-8610680, DMS8821330, and CES8657419.

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