ICES REPORT August T. Arbogast, D. Estep, B. Sheehan, and S. Tavener

Transcription

1 ICES REPORT 3-9 August 3 A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids by T. Arbogast, D. Estep, B. Sheehan, and S. Tavener The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 787 Reference: T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids, ICES REPORT 3-9, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, August 3.

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3 IMA Journal of Numerical Analysis (3) Page of 4 doi:.93/imanum/drnxxx A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids T. Arbogast, D. Estep, B. Sheehan and S. Tavener [Received on 3 August 3] The primary purpose of this paper is to compare the accuracy and performance of two numerical approaches to solving systems of partial differential equations. These equations are posed on adjoining domains sharing boundary conditions on a common boundary interface in the important case when the meshes used on the two domains are non-matching across the interface. The first widely used approach is based on a finite volume method employing ad hoc projections to relate approximations on the two domains across the interface. The second approach uses the mathematically-founded mortar mixed finite element method. To quantify the performance, we use a goal-oriented a-posteriori error estimate that quantifies various aspects of discretization error to the overall error. While the performance difference may be not a surprise in some cases, we believe that there is a perception in part of the scientific community concerned with multiphysics systems that if the solution is smooth near the interface, then there is little effect from varying the coupling technique. We find that, on the contrary, the error associated with ad hoc coupling approaches may be large in practical situations. Moreover, we also show that mortar methods can be used with black box component solves, thus permitting an efficient and practical implementation also within legacy codes. Keywords: mortar methods, a-posteriori error estimate, coupled elliptic problems, heterogeneous domain decomposition, geometric coupling. Introduction An important class of multiphysics problems has a structure in which one physical process dominates in one subdomain of the problem domain, while a second physical process dominates in a neighboring subdomain. The solutions are coupled by continuity of state and continuity of normal flux through a shared boundary between the subdomains. Examples include general problems of the heterogeneous domain decomposition type (Quarteroni et al., 99; Gaiffe et al., ; Bernardi et al., 994), core- Department of Mathematics, University of Texas at Austin (arbogast@math.utexas.edu). T. Arbogast s work was supported as part of the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC4. Department of Statistics, Colorado State University, Fort Collins, CO 853 (estep@stat.colostate.edu). D. Estep s work is supported in part by the Defense Threat Reduction Agency (HDTRA-9--36), Department of Energy (DE-FG- 4ER56, DE-FG-5ER5699, DE-FC-7ER5499, DE-SC74, DE-SC534, INL33), Idaho National Laboratory (6949, 5474), Lawrence Livermore National Laboratory (B584647, B59495), National Science Foundation (DMS-783, DMS-7535, DGE-5953, MSPA-CSE , ECCS-7559, DMS-6546, DMS-668, DMS-FRG-6546), National Institutes of Health (#RGM969). Department of Mathematics, Colorado State University (brendansheehan6@gmail.com). Department of Mathematics, Colorado State University. c The author 3. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

4 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER edge plasma simulations of a tokamak fusion experiment (Cary et al., 8, ), and conjugate heat transfer between a fluid and solid object (Estep et al., 8, 9b, ). In such situations, it is common to encounter significant differences in scales of behavior in the two subdomains. This in turn suggests the use of different discretization grids. However, this introduces the problem of interpreting the meaning of coupling state and flux values through the common boundary in the discretization, since exact pointwise matching is no longer possible. Confounding this issue are the practical difficulties of solving the large linear and nonlinear discrete systems associated with computing numerical solutions and the common situation in which two different codes are used to solve the two subdomain problems. These difficulties are generally tackled by employing some form of iterative approach that involves sequential solution of the subdomain problems. The particular properties of the discretizations used for each component problem, the choice of iterative solution method, and high performance computational considerations all have a large impact on the way in which state and flux values are passed across the common interface. In this paper, we investigate the accuracy of two approaches to computing the coupling values in the situation in which the discretization grids in the two subdomains do not match at the interface. The analysis is carried out for the closely related mixed finite element and cell-centered finite volume methods. The two approaches are () the mortar element approach (Brezzi & Fortin, 99; Roberts & Thomas, 99; Arbogast et al., ; Ben Belgacem, ; Arbogast et al., 7; Ganis & Yotov, 9), which uses a rigorous variational formulation to define a weak sense of coupling, and () a geometric approach that employs various ad hoc extrapolation and averaging methods. The use of mortar elements is proven to be optimally convergent on nonmatching grids, provided the finite element space used for the interface variables consists of piecewise polynomials of one degree higher than the trace along the interface of the finite element space used to approximate the flux within the subdomains (Arbogast et al., ). Nonetheless, while mortar elements are well known in some application domains, e.g., flow in porous media, they are not widely employed for multiphysics problems. Rather, various geometric techniques are used in most practical settings, especially in situations in which one or more of the components are solved with legacy black box codes. This second approach is often rationalized using a combination of ad hoc formal stability and/or accuracy arguments combined with high performance computing expediences. Moreover, in the situation in which legacy codes are used to solve either component, there is little choice because of the very considerable investment that would be required to replace these codes. We are not arguing for or against either mortar elements or geometric approaches. Rather, we address two issues: () What effect do these coupling approaches have on accuracy of specified quantities of interest? and () In each case, quantify the relative contributions of various aspects of discretization to the error in the computed information. The tool we use to address these issues is an adjoint-based a-posteriori error estimate (Estep et al., ; Becker & Rannacher, ; Giles & Suli, ; Wheeler & Yotov, 5; Estep et al., 9a; Hansbro & Larson, ; Pencheva et al., 3). This goal-oriented estimate accurately quantifies various contributions to the overall error. In particular, the estimate distinguishes contributions specifically arising from the mis-matched grids and the way in which the coupled information is approximated. We identify, through numerical examples, cases in which the geometric projections are the dominant source of error by one to two orders of magnitude. The remainder of this paper is organized as follows. Section two introduces the differential equation and the details of the two discrete methods. Section three derives the a-posteriori error estimate. Section four contains the numerical experiments. Section five discusses computational logistics related to iterative solvers, and a brief conclusion is given in section six.

5 ERROR ESTIMATES FOR MORTAR METHODS 3 of 4. Definition of the problem and discretization methods We define the coupled problem with a common interface, then describe the finite element and finite volume discretizations. We employ the well known equivalence between finite volume methods and the mixed finite element method (Russell & Wheeler, 983; Weiser & Wheeler, 988) to recast everything in the finite element framework. This greatly eases the derivation of a-posteriori error estimates and provides a systematic framework for describing geometric approaches to computing coupling values.. The differential equation The differential equation (.) (.3) consists of a system of second order elliptic partial differential equations (PDE) in two spatial dimensions. The system is posed on a rectangular domain Ω consisting of two nonoverlapping rectangular subdomains, Ω L on the left-hand side and Ω R on the right-hand side, that share a common interface Γ I, and whose union forms the entire domain, as shown in Fig.. The unit normal vector n is defined to point from left to right on Γ I, and is an outward pointing normal on Γ L = Ω L \ Γ I and Γ R = Ω R \ Γ I. For simplicity of presentation, we assume Dirichlet boundary conditions on Ω, the external boundaries of the domain. The results extend to problems with Neumann conditions on part of the boundary in a straightforward way. Γ L Γ R Γ L Ω L n Γ I Ω R Γ R Γ L Γ R FIG.. Subdomains, boundaries, and definition of normal n on the interface. For a diffusion function a, split as a L W, (Ω L ) and a R W, (Ω R ), source function f, split as f L L (Ω L ) and f R L (Ω R ), and boundary data g, split as g L H 3/ (Γ L ) and g R H 3/ (Γ R ), the coupled system is a L u L + p L =, (x,y) Ω L, u L = f L, (x,y) Ω L, (.) p L = g L, (x,y) Γ L, a R u R + p R =, (x,y) Ω R, u R = f R, (x,y) Ω R, (.) p R = g R, (x,y) Γ R, { ξ pl = p R, (x,y) Γ I, (.3) n (u L u R ) =, (x,y) Γ I,

6 4 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER where we assume that the diffusion matrices, a L and a R, are functions of space times the identity, i.e., [ DL (x,y) a L = D L (x,y) ] [ DR (x,y), a R = D R (x,y) ], (.4) with D i W, (Ω i ), i = L,R, and min D i (x,y) D >, so a i is invertible and uniformly coercive (x,y) Ω i for i = L,R. Note that we have defined ξ as the common interface pressure in (.3).. Mixed finite element mortar discretization The mortar finite element discretization was developed precisely for the situation presented by discretization of (.) (.3) using two different grids in the two different subdomains. We assume that each subdomain is discretized by a (logically) rectangular finite element grid. Lagrange multipliers are introduced on the interface boundary to provide a weak formulation of the pressure coupling conditions. Since the grids are different on the two sides of the interface, the Lagrange multiplier space cannot be the normal trace of the velocity space. So, we introduce a mortar finite element space on the interface (Arbogast et al., ; Bernardi et al., 5; Arbogast et al., 7). As shown in Arbogast et al. (), the method is optimally convergent and has several other desirable convergence properties if the boundary space has one order higher approximability than the normal trace of the velocity space. The same order of convergence is obtained for both continuous or discontinuous piecewise polynomials in the mortar space. In our discretization, we choose the interface grid that has one cell for every two cells in the finer of the two subdomain grids. Fig. shows the arrangement for a 5 5 grid next to 8 8 grid. (Note that our convention is that the finer grid is always used in the righthand subdomain.) FIG.. Example grid shown separated into the part on Ω L, Γ I, and Ω R from left to right. We use standard L inner product notation, i.e., for functions F and G defined on Ω, split as above, (F i,g i ) = F i (x,y)g i (x,y)dxdy, Ω i and for functions defined on the boundaries, we similarly denote F,G Γi = F Gds, Γ i i = L,I,R. i = L,R, The mixed finite element (mortar) method starts with the following continuous weak formulation. Find

7 ERROR ESTIMATES FOR MORTAR METHODS 5 of 4 p i W i = L (Ω i ), u i V i = H(div;Ω i ), ξ Λ = H / (Γ I ), i = L,R, satisfying (a L u L, v L ) (p L, v L ) + ξ, n v L ΓI = g L, n v L ΓL, ( u L,w L ) = ( f L,w L ), (a R u R, v R ) (p R, v R ) ξ, n v R ΓI = g R, n v R ΓR, ( u R,w R ) = ( f R,w R ), (.5) n (u L u R ),ν ΓI =, for all (w i, v i,ν) (W i,v i,λ), i = L,R. To discretize, we use the lowest order Raviart-Thomas finite element space (RT ), in which the discrete scalar unknown p h is approximated as a constant over each cell, and the components of the discrete vector u h are approximated by functions that are piecewise linear in one spatial dimension and constant in the other (Bernardi et al., 5; Estep et al., 9a). The discrete interface unknown, ξ h, is represented by piecewise discontinuous linears on the interface grid cells (Arbogast et al.,, 7). The test functions in the discretization of the weak formulation of (.5) corresponding to w, v, and ν are restricted to these same spaces. To be precise, for a finite element partition of [a,b], and for r =,,,..., q =,,,..., we define the piecewise polynomial space M r q ( ) = {v C q ([a,b]) : v is a polynomial of degree r on each subinterval of }. When q = the functions are discontinuous. The space of continuous piecewise bilinear functions is the tensor product M ( x) M ( y). The RT discrete spaces are W h i = M ( x,i) M ( y,i), i = L,R, V h i = [M ( x,i ) M ( y,i)] [M ( x,i) M ( y,i )], Λ h = M ( ΓI ). i = L,R, The mixed finite element (mortar) method reads: Compute p h i W h i, uh i V h i, ξ h Λ h, i = L,R, satisfying (a L uh L, v L ) (p h L, v L ) + ξ h, n v L ΓI = g L, n v L ΓL, ( u h L,w L ) = ( f L,w L ), (a R uh R, v R ) (p h R, v R ) ξ h, n v R ΓI = g R, n v R ΓR, ( u h R,w R ) = ( f R,w R ), (.6) n (u h L u h R),ν ΓI =, for all (w i, v i,ν) (Wi h,v h i,λ h ), i = L,R. This yields a discrete system of the form M L B L C L u h B T L D L L p h L M R B R C R B T u h R R p h = F L D R CL T CR T R F R, (.7) ξ h where we abuse notation to let u h i, ph i, and ξ h denote the vector of nodal values for the finite element functions.

8 6 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER.3 Geometrically coupled finite volume discretization The standard formulation of the finite volume method eschews a variational formulation of the problem, so there is no natural description of a weak imposition of the coupling conditions in that formulation. Moreover, the standard finite volume method provides approximation values of p only at cell centers while approximate values for u along cell boundaries are obtained by differencing the p values. These characteristics motivate the use of geometric coupling techniques that employ a combination of extrapolation and averaging to provide coupling values of both unknowns along the interface. The motivation for this approach is reinforced in the context of iterative solution of the coupled problems, where well posed problems are created on each subdomain using interface boundary conditions obtained from the other subdomain. In this approach, it is necessary to couple the coarser side using state values extrapolated from the finer side solution, while the finer side must be coupled to flux values, which are themselves differences of state values, extrapolated from the coarser solution. Reversing this arrangement can lead to a singular system. D E A B C FIG. 3. Extrapolation to the interface. Left: Neumann values on the interface are computed by linear extrapolation of the last two available flux values, which are differences of state values. Right: Dirichlet values on the interface are computed by linear extrapolation of the last two available state values. FIG. 4. Averaging or broadcasting of extrapolated values. Left: In the case of constant extrapolation, the last available state or flux value is simply used as the interface value. Right: Weighted averaging of state and flux values when cell widths do not share an integer ratio. To obtain values on the interface, we employ either linear or constant extrapolation. We illustrate linear extrapolation in Fig. 3. We compute the extrapolated values by computing a linear or constant interpolant, which is then evaluated at the interface boundary. We denote the extrapolated values using the operators P R L (p h R ) and P L R(p h L ). When the cells on either side of the interface do not match, then weighted averaging and broadcasting schemes are used to generate values. In Fig. 4, we illustrate the averaging and broadcasting schemes when two cells on the right match one cell on the left. The state values at the two circle locations are averaged and used at the square location. The flux value at the square location is broadcast to both of the circle locations. When the cell widths on the coarse and fine side of the interface do not share an integer ratio, then a suitable averaging of values is used. For example, in the cells next to 3 cells arrangement pictured in Fig. 4, the state value at location D is set equal to 3 the state value at location A plus 3 the state value at location B. The flux value at location A is set equal to the flux value at location D, while the flux value used at location B is set equal to half the flux value at D plus half the flux value at E. We formulate the finite volume method as an RT mixed finite element method employing a special quadrature formula, following Russell & Wheeler (983); Weiser & Wheeler (988). This provides a foundation for deriving an a-posteriori error analysis for the finite volume scheme, see Estep et al.

9 ERROR ESTIMATES FOR MORTAR METHODS 7 of 4 (9a). The version of (.6) equivalent to a finite volume method reads: Compute p h i W h i, uh i V h i, ξ h Λ h, i = L,R, satisfying (a L uh L, v L ) M,T (p h L, v L ) + P R L (p h R), n v L ΓI = g L, n v L ΓL,M, ( u hl,w L ) = ( f L,w L ), (a R uh R, v R ) M,T (p h R, v R ) ξ h, n v R ΓI = g R, n v R ΓR,M, ( u h R,w R ) = ( f R,w R ), (.8) (P L R (p h L) n u h R),ν ΓI =, for all (w i, v i,ν) (W h i,v h i,λ h ), i = L,R. Here we employ the approximate inner product (u h, v) M,T = (u h x,v x ) Tx,M y + (u h y,v y ) Mx,T y, where M ( ) and T ( ) denote the midpoint and trapezoidal quadrature rules in the x and y directions as indicated, while, Γi,M denotes the midpoint rule for i = L,R. Note the quadrature formulas are applied internally on each cell, so potential discontinuites in a and f across Γ I cause no difficulty. This yields a discrete system of the form M L B L Q D B T L M R B R C R B T R Q N CR T u h L p h L u h R p h R ξ h = D L F L D R F R, (.9) which should be compared to (.7). It is possible to eliminate the unknowns u h i, i = L,R, and ξ h, to reduce (.9) to a system for p h i of the form [ ][ ] [ ] AL C D p h L FL C N A R p h =. (.) R F R The averaging and broadcasting are incorporated into the coupling Dirichlet and coupling Neumann matrices C D and C N. This is the same system that is constructed by using a finite volume approach directly. We have verified through numerical experiments that the p component of the solution of (.9) is identical to the solution of (.). Furthermore, the u component of the solution of (.9) is identical to the u values obtained by differencing the solution of (.) to approximate p at the cell boundaries and evaluating the diffusivity at the cell boundaries. The ξ component of the solution of (.9) has no counterpart in the solution of (.). 3. A-posteriori error analysis Our goal is to derive an a-posteriori error estimate for the quantity of interest (e ul, ψ ul ) + (e pl,ψ pl ) + (e ur, ψ ur ) + (e pr,ψ pr ) + e ξ,ψ ξ ΓI, (3.) where ψ ul,ψ pl, ψ ur,ψ pr, and ψ ξ are given L functions and e ( ) denotes the errors in the corresponding variables. We define the generalized Green s function corresponding to these functionals using the

10 8 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER adjoint problem a L φ L ζ L = ψ ul on Ω L, φ L = ψ pl on Ω L, ζ L = on Γ L, a R φ R ζ R = ψ ur on Ω R, φ R = ψ pr on Ω R, ζ R = on Γ R, { β ζl = ζ R on Γ I, n (φ L φ R ) = ψ ξ on Γ I. (3.) (3.3) (3.4) The a-posteriori error estimates explicitly depend on φ L,ζ L, φ R, and ζ R. 3. Estimate for mortar mixed finite element method We first derive an estimate for the mortar finite element method assuming all integrals in the weak formulation are computed exactly. We begin by substituting (3.) (3.4) for the various ψ s in (3.) and applying the divergence theorem, Expanding on the right and subtracting (e ul, ψ ul ) + (e pl,ψ pl ) + (e ur, ψ ur ) + (e pr,ψ pr ) + e ξ,ψ ξ ΓI (3.5) = (e ul,a L φ L ) + ( e ul,ζ L ) n e ul,β ΓI (e pl, φ L ) (e ur,a R φ R ) + ( e ur,ζ R ) + n e ur,β ΓI (e pr, φ R ) + e ξ, n (φ L φ R ) ΓI. (a L u L, φ L ) (p L, φ L ) + ξ, n φ L ΓI + g L, n φ L ΓL +( u L,ζ L ) ( f L,ζ L ) +(a R u R, φ R ) (p R, φ R ) ξ, n φ R ΓI + g R, n φ R ΓR +( u R,ζ R ) ( f R,ζ R ) n (u L u R ),β ΓI =, obtained by substituting the adjoint solution as test functions into the forward weak form (.5), gives (e ul, ψ ul ) + (e pl,ψ pl ) + (e ur, ψ ur ) + (e pr,ψ pr ) + e ξ,ψ ξ ΓI (3.6) = (a L uh L, φ L ) + (p h L, φ L ) g L, n φ L ΓL ξ h, n φ L ΓI +( f L,ζ L ) ( u h L,ζ L ) (a R uh R, φ R ) + (p h R, φ R ) g R, n φ R ΓR + ξ h, n φ R ΓI +( f R,ζ R ) ( u h R,ζ R ) + n (u h L u h R),β ΓI.

11 ERROR ESTIMATES FOR MORTAR METHODS 9 of 4 We rewrite this as wherein the residuals are given by Note that the divergence theorem implies (e ul, ψ ul ) + (e pl,ψ pl ) + (e ur, ψ ur ) + (e pr,ψ pr ) + e ξ,ψ ξ ΓI (3.7) = (R ul, φ L ) + (R pl,ζ L ) + (R ur, φ R ) + (R pr,ζ R ) + R ξ,β ΓI, R ul = a L uh L p h L, R ur = a R uh R p h R, R pl = f L u h L, R pr = f R u h R, R ξ = n (u h L u h R). (R ul, φ L ) = (a L uh L, φ L ) + (p h L, φ L ) p h L, n φ L ΩL = (a L u hl, φ L ) + (p h L, φ L ) g L, n φ L ΓL ξ h, n φ L ΓI. Also note that β = ζ L = ζ R for the continuous adjoint solution, but β is distinct from ζ L and ζ R for the discrete solution. Next, we use Galerkin orthogonality. We introduce projection operators that map into the finite element space of the discrete forward solution: P h L : L (Ω L ) W h L, P h R : L (Ω R ) W h R, Π h L : L (Ω L ) V h L, Π h R : L (Ω R ) V h R, Z h : L (Γ I ) Λ h. The actual choice of projection is immaterial for the estimate. In practice, we employ a combination of restriction and averaging. Without quadrature, Galerkin orthogonality for (.6) is expressed as (R ul,π h L φ L ) + (R pl,p h L ζ L ) + (R ur,π h Rφ R ) + (R pr,p h Rζ R ) + R ξ,z h β ΓI =, and subtracting gives the following result. THEOREM 3. The errors for the mixed finite element method (.6) without quadrature satisfy (e pl,ψ pl ) + (e ul, ψ ul ) + (e pr,ψ pr ) + (e ur, ψ ur ) + e ξ,ψ ξ ΓI (3.8) = (R ul, φ L Π h L φ L ) + (R pl,ζ L P h L ζ L ) + (R ur, φ R Π h Rφ R ) + (R pr,ζ R P h Rζ R ) + R ξ,β Z h β ΓI, wherein the quantities on the right-hand side are computable provided the true adjoint solution is available. In practice, we employ a numerical solution of the adjoint problem. To emphasize this, we state the following corollary that involves numerical adjoint quantities. COROLLARY 3. Provided that the projection operators P h L, Ph R, Π h L, Π h R, and Z h are bounded in L, the errors for the mixed finite element method (.6) without quadrature can be estimated as (e pl,ψ pl ) + (e ul, ψ ul ) + (e pr,ψ pr ) + (e ur, ψ ur ) + e ξ,ψ ξ ΓI (3.9) (R ul, φ h L Π h L φ h L ) + (R p L,ζ h L P h L ζ h L ) + (R ur, φ h R Π h Rφ h R ) + (R p R,ζ h R P h Rζ h R) + R ξ,β h Z h β h ΓI, for numerical solutions φ L φ h L, ζ L ζ h L, φ R φ h R, ζ R ζ h R, and β β h. In this approximation, the errors are to be measured in the L -norm.

12 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER The proof follows from the triangle inequality and the definition of the operator norm. That is, the absolute value of the difference between the right-hand sides of (3.8) and (3.9) is bounded by ( + Π h L ) R ul φ L φ h L + ( + P h L ) R pl ζ L ζ h L + ( + Π h R ) R ur φ R φ h R + ( + P h R ) R pr ζ R ζ h R + ( + Z h ) R ξ,γi β β h,γi. In order to obtain accurate estimates, the numerical adjoint solutions must be sufficiently accurate. Generally this is satisfied by solving the adjoint problems either using a higher order numerical method or using a mesh sufficiently refined from the one used for the forward discretization. In the context of finite volume discretizations, the second approach is generally easier to implement. In our numerical examples we use a finer grid, and the accuracy of this approach is illustrated in section Estimate for finite volume methods using geometric coupling 3.. The effect of quadrature. We first derive an estimate for the mixed finite element method (.6) with quadrature, which can be applied, say, if f i, g i, and a i are continuous in Ω i, i = L,R. With quadrature, Galerkin orthogonality is expressed as (R ul,π h L φ L ) Q + (R pl,p h L ζ L ) Q + (R ur,π h Rφ R ) Q + (R pr,p h Rζ R ) Q + R ξ,z h β Q,ΓI =, where we use the subscript Q to denote the approximate inner product using quadrature. It is important to distinguish residuals associated with approximating the solution spaces using finite dimensional polynomial spaces from residuals associated with approximating the integrals defining the variational formulation. We rewrite Galerkin orthogonality as with (R ul,π h L φ L ) + (R pl,p h L ζ L ) + (R ur,π h Rφ R ) + (R pr,p h Rζ R ) + R ξ,z h β ΓI QE ul (Π h L φ L ) QE pl (P h L ζ L ) QE ur (Π h Rφ R ) QE pr (P h Rζ R ) QE ξ (Z h β) =, QE ul (Π h L φ L ) = (R ul,π h L φ L ) (R ul,π h L φ L ) Q, QE pl (P h L ζ L ) = (R pl,p h L ζ L ) (R pl,p h L ζ L ) Q, QE ur (Π h Rφ R ) = (R ur,π h Rφ R ) (R ur,π h Rφ R ) Q, QE pr (P h Rζ R ) = (R pr,p h Rζ R ) (R pr,p h Rζ R ) Q, QE ξ (Z h β) = R ξ,z h β ΓI R ξ,z h β Q,ΓI. This gives the following a-posteriori estimate for the mixed finite element method with quadrature. THEOREM 3. If f i, g i, and a i are continuous in Ω i, i = L,R, then the errors for the mixed finite element method (.6) with quadrature satisfy (e pl,ψ pl ) + (e ul, ψ ul ) + (e pr,ψ pr ) + (e ur, ψ ur ) + e ξ,ψ ξ ΓI (3.) = (R ul, φ L Π h L φ L ) + (R pl,ζ L P h L ζ L ) + (R ur, φ R Π h Rφ R ) + (R pr,ζ R P h Rζ R ) + R ξ,β Z h β ΓI + QE ul (Π h L φ L ) + QE pl (P h L ζ L ) + QE ur (Π h Rφ R ) + QE pr (P h Rζ R ) + QE ξ (Z h β).

13 ERROR ESTIMATES FOR MORTAR METHODS of 4 Note that in the case of using the RT finite element space and the midpoint-trapezoidal quadrature rules discussed above, the mixed finite element method reduces to the finite volume method (Russell & Wheeler, 983; Weiser & Wheeler, 988; Estep et al., 9a), and some of the quadrature error terms are zero. These terms are included for generality, so that (3.) is valid for other combinations of finite element spaces and quadratures. Note that in practice, we implement the obvious analog of Corollary 3., which now requires sufficient smoothness of the solution to obtain sufficiently accurate quadrature approximations. 3.. The effect of geometric coupling. For the geometric coupling (.8), the Galerkin orthogonality becomes (R ul,π h L φ L ) Q P R L (p h R) ξ h, n Π h L φ L Q,ΓI + (R pl,p h L ζ L ) Q Defining + (R ur,π h Rφ R ) Q + (R pr,p h Rζ R ) Q + R ξ,z h β Q,ΓI n u h L P L R (p h L),Z h β Q,ΓI =. QE ul (Π h L φ L ) = (R ul,π h L φ L ) (R ul,π h L φ L ) Q + P R L (p h R) ξ h,π h L φ L Q,ΓI P R L (p h R) ξ h, n Π h L φ L ΓI, QE ξ (Z h φ ) = R ξ,z h β ΓI R ξ,z h β Q,ΓI and arguing as above gives the following result. + n L u h L P L R (p h L),Z h β Q,ΓI n u h L P L R (p h L),Z h β ΓI, THEOREM 3.3 If f i, g i, and a i are continuous in Ω i, i = L,R, then the error for the mixed geometric finite volume method (.8) satisfies (e pl,ψ pl ) + (e ul, ψ ul ) + (e pr,ψ pr ) + (e ur, ψ ur ) + e ξ,ψ ξ ΓI (3.) = (R ul, φ L Π h L φ L ) + P R L (p h R) ξ h, n Π h L φ L ΓI + (R pl,ζ L P h L ζ L ) + (R ur, φ R Π h Rφ R ) + (R pr,ζ R P h Rζ R ) + R ξ,β Z h β ΓI + n u h L P L R (p h L),Z h β ΓI + QE ul (Π h L φ L ) + QE pl (P h L ζ L ) + QE ur (Π h Rφ R ) + QE pr (P h Rζ R ) + QE ξ (Z h β). Note that in practice, we implement the obvious analog of Corollary 3., assuming again sufficient smoothness of the solution to obtain sufficiently accurate quadrature approximations. 4. Numerical investigations In this section, we use the a-posteriori error estimates to investigate in detail the accuracy of the two approaches to coupling. For all of the investigations, the coarser subdomain Ω L is given by x [,] and y [,], the finer subdomain Ω R is given by x [,] and y [,] (see Fig. ), and the interface Γ I is located along y =. (Note that here the bottom subdomain is considered as being left and the top one is right, in conformance to our convention as to the finer subdomain.) The grids are reported as n L m L for the left domain and n R m R for the right domain, where n ( ) corresponds to the number of cells in the x-direction (which is also the number of cells along the interface), and m ( )

14 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER corresponds to the number of cells in the y-direction. The boundary conditions for all tests are Dirichlet. To avoid issues arising from iterative solution of the discrete system, we employ direct methods to find the approximate solution to within machine precision. The quantity of interest being sought is specified by giving the adjoint problem data ψ ux, ψ uy, ψ p, and ψ ξ. The adjoint problem is solved using the same RT mixed finite element method, but on a grid that is significantly finer than that of the forward problem, so that the discretization error associated with the adjoint solution has no significant effect on the results. The functions chosen for the source, diffusivity, and adjoint data are either constants or Gaussian functions of the form ae (y b) + K, c which gives a localized ridge centered at y = b. In the case of the adjoint data, the Gaussian or constant function being used is normalized so that the area under ψ is equal to one. The parameter K is non zero only in the case of diffusivity, where this constant is added to the Gaussian to prevent the diffusivity from approaching zero anywhere in the domain. In the tests, we report values for the terms in (3.) and (3.) that are non zero. For both the mixed finite element and geometric finite volume methods the following five terms are included: MFE or GFV = (R ul, φ h L Π h L φ h L ), MFE or GFV = (R ur, φ h R Π h Rφ h R ), MFE 3 or GFV 3 = (R pl,ζ h L P h L ζ h L ), MFE 4 or GFV 4 = (R pr,ζ h R P h Rζ h R), MFE 5 or GFV 5 = R ξ,β h Z h β h ΓI. In the geometric finite volume case, we add two additional terms relating to the geometric projections and two additional quadrature terms: GFV 6 = P R L (p h R) ξ h, n Π h L φ h L Γ I, GFV 7 = n u h L P L R (p h L),Z h β h ΓI, GFV 8 = QE ul (Π h L φ h L ), GFV 9 = QE ur (Π h Rφ h R ). We note that the first five expressions, common to both MFE and GFV, are often similar in size. As a gross measure of the effect of geometric projection and of the use of quadrature, we also report the two ratios ratio proj = 7 i=6 GFV i 5 i= GFV i, ratio quad = 9 i=8 GFV i 5 i= GFV i. We present three examples chosen to show the spectrum of possibilities in terms of the performance of the methods. Test case is an easy case in which the geometric method performs relatively well. Test case has a narrow and severe dip in diffusivity located along the subdomain boundary and consquently the geometric method performs poorly, as might be expected for a problem in which the solution changes rapidly near the interface. Test case 3 is based loosely on a real world fusion problem and demonstrates one of our main conclusions, which is that the geometric method can perform poorly

15 ERROR ESTIMATES FOR MORTAR METHODS 3 of 4 even when the solution is smooth near the interface. Note that the behavior of the diffusivity function approaching the common interface on both sides has more impact on the accuracy of coupling than a discontinuity in the diffusivity across the interface. 4. Verification of a-posteriori estimate accuracy We begin with a problem for which we have manufactured the known solution ( πx ) ( πy ) p(x,y) = cos cos. (4.) 4 The diffusivity a is equal to one everywhere. The other solution components, the source term f, and the boundary values g for the problem follow from (4.). Since we know the true solution, we can compute the exact error terms (e,ψ) on the left in (3.) and (3.) directly and then compare to estimates of the quantities on the right computed using a numerical solution to the adjoint problem. In this situation, the most important issue for the accuracy of the estimates is the accuracy of the approximate adjoint solutions. As the grid for the adjoint problem is refined, the estimates become more accurate. That is, using the approximation to the adjoint problem, the estimated quantities MFE i or GFV i becomes closer to their true value, the error in the quantity of interest MFE (e,ψ) or GFV (e,ψ). Tables and show this using coarse and fine forward solutions. Table. The forward problem with solution (4.) is run at next to 6 6. The adjoint problem is run at several grids to show how the sum of terms approaches the direct calculation of (e,ψ). The adjoint data components ψ u and ψ p are constant everywhere and ψ ξ =. adj. grid MFE (e,ψ) MFE i ratio GFV (e,ψ) GFV i ratio x : 3x3.96E 3.47E E 3.5E x4 : 64x64.96E 3.84E E 3.3E 3. 8x8 : 8x8.96E 3.93E E 3.3E 3.3 6x6 : 56x56.96E 3.96E E 3.E 3. Table. The forward problem with solution (4.) is run at 4 4 next to The adjoint problem is run at several grids to show how the sum of terms approaches the direct calculation of (e,ψ). The adjoint data components ψ u and ψ p are constant everywhere and ψ ξ =. adj. grid MFE (e,ψ) MFE i ratio GFV (e,ψ) GFV i ratio 8x8 : 8x8.3E 4 9.3E E 5.E x6 : 56x56.3E 4.5E E E Convergence To compare the accuracy of the various approximations, we use the -norms e p = (p p h ), e ux = (u x u h x), Ω Ω e uy = (u y u h y), e ξ = (ξ ξ h ). Ω Γ

16 4 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER We use the manufactured solution from the previous section (a = and p is given by (4.)). We compare the -norm errors of the finite element and geometric finite volume methods on a sequence of grids in order to asses the convergence rate. The coarsest grid is next to 6 6, and the number of cells in each dimension is doubled with each refinement. The results in Tables 3 6 show that the convergence rate for the geometric finite volume deteriorates for the u x, u y, and ξ components when the number of cells along the fine side of the interface is not an integer multiple of the number of cells along the coarse side of the interface. When the test is repeated with a grid starting at 8 8 next to 6 6, the convergence rates for the two methods are equal. The first order convergence of p and u for the MFE is to be expected (Arbogast et al., ). Table 3. Convergence of solution component p, indicating a rate of O(h). grid MFE e p MFE ratio GFV e p GFV ratio x : 6x6.E- N/A.E- N/A x : 3x3 5.98E E-. 4x4 : 64x64.99E-..99E-. 8x8 : 8x8.49E-..49E-. 6x6 : 56x E E-3. Table 4. Convergence of solution component u x, indicating a rate of about O(h). grid MFE e ux MFE ratio GFV e ux GFV ratio x : 6x6 8.49E- N/A 8.63E- N/A x : 3x3 4.E-. 4.6E-. 4x4 : 64x64.E-..4E-.99 8x8 : 8x8.5E-..9E-.97 6x6 : 56x56 5.5E E-3.93 Table 5. Convergence of solution component u y, indicating a rate of O(h) for MFE but less for GFV. grid MFE e uy MFE ratio GFV e uy GFV ratio x : 6x6 8.4E- N/A 8.59E- N/A x : 3x3 4.E E-.96 4x4 : 64x64.E-..9E-.9 8x8 : 8x8.5E-..3E-.86 6x6 : 56x56 5.5E E-3.77 Table 6. Convergence of solution component ξ, indicating a rate of O(h ) for MFE but only O(h) for GFV. grid MFE e ξ MFE ratio GFV e ξ GFV ratio x : 6x6 7.53e-3 N/A 6.e-3 N/A x : 3x3.89e e x4 : 64x64 4.7e e-4.8 8x8 : 8x8.8e e-4.3 6x6 : 56x56.95e e-4.9

17 4.3 Test Case ERROR ESTIMATES FOR MORTAR METHODS 5 of 4 In the next problem, we explore accuracy for a solution that is not changing rapidly near the interface. We find that the use of geometric projections does not lead to significant effects on accuracy. We let the diffusivity a be one in both Ω L and Ω R and use the manufactured solution given by (4.). The grid for the forward problem is next to 3 3. The adjoint grid is 8 8 next to 8 8, and the adjoint data is a nonzero constant for ψ ux, ψ uy, and ψ p, while ψ ξ =. We list the error contributions in Table 7. For the geometric approach, we list results for both constant and linear extrapolation. The results show that the projection error for linear extrapolation is only about one quarter of the residual error, while the projection error for constant extrapolation is much larger. Fig. 5 shows the solution components for the finite element case. The geometric finite volume solutions are very similar. Fig. 6 shows the adjoint solution components. Table 7. Error terms for Case. The forward grid is next to 3 3. The adjoint grid is 8 8 next to 8 8. term MFE GFV (linear) GFV (constant) (R ul, φ h L Π L hφ h L ).6E 4.6E 4.5E 4 (R ur, φ h R Π R hφ h R ) 6.E 5 6.E 5 6.E 5 3 (R pl,ζl h Ph L ζ L h ) 4.9E 4 4.9E 4 4.9E 4 4 (R pr,ζr h Ph R ζ R h ).9E 4.9E 4.9E 4 5 R ξ,β h Z h β h ΓI 4.E 8.3E 6.E 5 6 P R L (p h R ) ξ h, n ΠL hφ h L Γ I N/A.E 4.7E 4 7 n u h L P L R(p h L ),Zh β h ΓI N/A.E 5 3.9E 3 8 QE ul (ΠL hφ h L ) N/A 7.E 4 7.E 4 9 QE ur (ΠR hφ h R ) N/A.8E 4.7E 4 total 4.6E 4 3.E 4 3.6E 3 ratio proj N/A ratio quad N/A p h u h x u h y ξ h FIG. 5. Finite element solution components for Case.

18 6 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER ζ h φ h x φ h y β h FIG. 6. Adjoint solution components for Case. Shown are plots for an adjoint solution using a 4 4 grid next to grid. A solution on a finer grid is used to compute the estimates. 4.4 Test Case The next test problem presents a more difficult solution for which the geometric projection error is by far the largest source of error. The grid is 4 4 next to and the boundary conditions are g = on both subdomains. Fig. 7 shows profiles of the source and diffusivity, while Fig. 8 shows the adjoint data f (y) a(y) FIG. 7. Source f (left) and diffusivity a (right) profiles for Test Case. The plots are shown in one dimension since the source and diffusivity have no variation in the x-direction ψ ux (y) ψ uy (y) ψ p (y) ψ ξ FIG. 8. Adjoint data profiles for Case. The plots of ψ ux, ψ uy, and ψ p are shown in one dimension because they have no variation in the x-direction. The shapes in Fig. 7 and 8 are based on a normalized gaussian of the form ae (x b) c. The parameter a is for normalization and is set to a = c. The parameter b determiness the location of the peak and is set to zero to coincide with the interaface. The π parameter c determines the width of the peak and is set to c =.. In the case of diffusivity the dip is produced using the function.3 ( ae (x b) c ).

19 7 of 4 ERROR ESTIMATES FOR MORTAR METHODS Because the source is large but the diffusivity is small along the interface, the solution changes rapidly near this region. This leads to relatively large errors near the interface for the geometric finite volume method. When the adjoint data is concentrated near the interface, the relative size of these errors is revealed. Table 8 lists the error terms. For this particular example problem, and this particular error measure, the error due to geometric projection is nearly eighty times the total error associated with the residuals. Fig. 9 shows the solution components for the finite element case, Fig. shows the solution components for the geometric finite volume case, and Fig. shows the adjoint solution. Table 8. Error terms for Case. The forward grid is 4 4 next to The adjoint grid is 6 6 next to term Ru L, φ hl ΠLh φ hl ) (R Ru R, φ hr ΠRh φ hr ) (R (R pl, ζlh PLh ζlh ) (R pr, ζrh PRh ζrh ) hrξ, β h Zh β h iγi hpr L (phr ) ξ h, n ΠLh φ hl iγi hnn u hl PL R (phl ), Z h β h iγi QEu L (ΠLh φ hl ) QEu L (ΠLh φ hl ) total ratioproj ratioquad GFV (linear).9e 5.6E 5 3.E 5 4.6E 5 9.4E 6.3E 3 9.E 4.E 3.4E 4 4.E 3 5 MFE.6E 5.6E 5 3.E 5 4.6E 5 4.8E 8 N/A N/A N/A N/A 5.E 6 N/A N/A GFV (constant).4e 5.5E 5 3.E 5 4.6E 5.6E 5.7E 3 8.4E 3.E 3.5E 4.E ph uhx uhy ξh FIG. 9. Finite element solution components for Case. Zooming in reveals that uhy is smooth and continuous across the interface ph uhx uhy ξh FIG.. Geometric finite volume solution components for Case. Zooming in reveals that uhy is discontinuous across the interface.

20 8 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER ζ h φ h x φ h y β h FIG.. Adjoint solution components for Case. Shown are plots for an adjoint solution using a 4 4 grid next to grid. A solution on a finer grid is used to compute the estimates. 4.5 Test Case 3 In our final example, we examine a problem that places only one cell in the x-direction in one of the subdomains. Such a grid is only appropriate if the solution in that subdomain is essentially one dimensional, and varies only parallel to the interface. This situation arises in core-edge coupling in a tokamak fusion reactor. We construct a problem with a solution that is very nearly one dimensional in one subdomain, and contains variation in the second dimension well away from the interface. The pressure component of the solution is ( π(y + ) p(x,y) = cos 8 ) +.3 sin(πx) [ tanh ( (.5 y) ) ]. (4.) The grid is 3 next to 3 3 and the boundary conditions are provided by evaluating the known solution at the outer domain boundaries. The source for the problem is computed by substituting the chosen solution into the PDE. The diffusivity a is one in both Ω L and Ω R. The adjoint data is concentrated in the finer subdomain, and is shown in Fig ψ ux (y) ψ uy (y) ψ p (y) ψ ξ FIG.. Adjoint data profiles for Case 3. The plots of ψ ux, ψ uy, and ψ p are shown in one dimension because they have no variation in the x-direction, and ψ ξ is a one dimensional function defined on the interface. Table 9 lists the error terms. For this example problem, the contribution due to geometric projection with linear extrapolation is approximately ten times the total contribution associated with the residuals, despite the fact that the solution is changing slowly near the interface. The projection contribution is much larger if constant extrapolation is used. Fig. 3 shows the solution components for the finite element case, Fig. 4 shows the solution components for the geometric finite volume case, and Fig. 5 shows the adjoint solution components.

21 ERROR ESTIMATES FOR MORTAR METHODS 9 of 4 Table 9. Error terms for Case 3. The forward grid is 3 next to 3 3. The adjoint grid is 8 8 next to 8 8. term MFE GFV (linear) GFV (constant) (R ul, φ h L Π L hφ h L ) 3.9E 9 6.8E 7.5E 5 (R ur, φ h R Π R hφ h R ).8E 5.8E 5.8E 5 3 (R pl,ζl h Ph L ζ L h ) 4.E 6 4.E 6 4.E 6 4 (R pr,ζr h Ph R ζ R h ) 7.E 6 7.E 6 7.E 6 5 R ξ,β h Z h β h ΓI 3.8E 7.7E 5 6 P R L (p h R ) ξ h, n ΠL hφ h L Γ I N/A.5E 4 6.4E 3 7 n u h L P L R(p h L ),Zh β h ΓI N/A 6.8E 5 3.E 3 8 QE ul (ΠL hφ h L ) N/A.4E 5.5E 3 9 QE ur (ΠR hφ h R ) N/A.E 5.9E 5 total.e 5.E 4 8.E 4 ratio proj N/A ratio quad N/A p h u h x u h y ξ h FIG. 3. Finite element solution components for Case p h u h x u h y ξ h FIG. 4. Geometric finite volume solution components for Case 3 computed using linear extrapolation.

22 of 4 ARBOGAST, ESTEP, SHEEHAN, TAVENER ζ h φ h x φ h y β h FIG. 5. Adjoint solution components for Case 3. These plots are the adjoint solution using next to meshes. The estimates were computing using a finer grid. 5. Iterative solvers and coupling strategies In practice, iterative solution of the coupled system is often employed. The specific choice of solution method is often constrained by certain computational logistics, such as the state of existing codes and data structures. We briefly discuss some aspects of iterative solution. The primary goal is to show that iterative solution strategies applied to systems like (.) can also be applied to systems like (.7) without large changes to the computational structure. We do not discuss the convergence of iterative solvers. 5. Iteration on the primary variable A common iterative technique for the geometric finite volume method (.) is to start with an initial guess (p L, p R ) and proceed with the iteration [ ] [ ] [ ] AL FL CD =, i =,,,... (5.) A R F R C N ][ p i+ L p i+ R ][ p i L p i R This iteration requires only the inversion of A L and A R, that is, only single domain component solves. The application of C D and C N can be viewed as the coupling strategy, in which information is swapped between the subdomains. It is possible to use an iteration of this type on the finite element system (.7) as well. We must first reduce to a system in p by a preprocessing procedure. We first eliminate u L and u R, which results in B T L M L B L B T L M L B T R M R B R B T R M R C R CL T M L B L CR T M R B R (CL T M L C L +CR T M R C R) C L p L p R ξ = F L + B T L M L D L F R + B T R M R D R D L +CR T M C T L M L R D R, (5.) which we write succinctly as G L H L G R H R HL T HR T (K L + K R ) p L p R ξ = R L R R S L + S R. (5.3)

23 We then eliminate ξ to obtain ERROR ESTIMATES FOR MORTAR METHODS of 4 [ GL H L (K L + K R ) H T L H L (K L + K R ) H T R ][ pl ] H R (K L + K R ) H T L G R H R (K L + K R ) H T R = p R [ RL H L (K L + K R ) (S L + S R ) R R H R (K L + K R ) (S L + S R ) ]. (5.4) System (5.4) has the same structure as (.), so an iteration analogous to (5.) can be applied. The stencil within the diagonal blocks of (5.4) is very close, but not identical, to the stencil of a single domain discretization. The difference occurs only in the stencil corresponding to cells touching the interface. In some cases, e.g., the use of black box single domain solvers, it is necessary to construct a system in which the diagonal blocks correspond exactly to single domain discretizations. If this is the case, the strategy of discretization consistent interface conditions provides a partial solution. In this strategy, the diagonal blocks are single domain discretizations, just as in (.). The off diagonal blocks are populated by writing down both the Dirichlet and Neumann boundary condition equations for every cell touching the interface, rearranging those equations to isolate the boundary value terms and setting those terms equal to each other across the interface. If the cell ratio along the interface is integer, such as 4 next to 8, the resulting system is algebraically equivalent to (5.4). If the cell ratio is not an integer ratio, such as 5 next to 8, the equality of boundary value terms across the interface can only be enforced approximately, and the resulting system is not exactly equivalent to (5.4). While a complete discussion of the implementation of discretization consistent interface conditions is beyond the scope of this paper, it is worth consideration as an alternative to the full mortar method in cases where the computational structure is constrained by black box single domain solvers in combination with iteration on the primary variables. The concept of discretization consistent interface conditions is similar to strategies employed in Farhat et al. (998) and Edwards & Rogers (998). We should remark that the former paper recommended against mortar methods for the fluid-structure interaction problem, due to the lack of theory on optimal convergence and a need to invert a large interface matrix. However, for the problem considered in this paper, the mortar method does achieve optimal convergence. Moreover, we presented several computational strategies that do not require inversion of an interface matrix. 5. Iteration on interface variables An alternative iterative strategy (Glowinski & Wheeler, 988) uses the interface variables as the primary variables. If we combine the u and p variables into the symbol ψ, then system (.7) can be written as We eliminate ψ as which gives the following system for ξ : (C T L A L A L C L A R C R CL T CR T ψ L ψ R ξ = F L F R ψ i = A i (F i C i ξ ), i = L,R,. (5.5) C L + CR T A C R)ξ = (CL T A F L + CR T AR F R). (5.6) R L