Mathematical Modelling and Numerical Analysis SAIM: MAN Modélisation Mathématique et Analyse Numérique MAN, Vol. 37, N o, 3, pp. 9 5 DOI:.5/man:33 A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS Roland Becker, Peter Hansbo and Rolf Stenberg 3 Abstract. In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (97) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included. Mathematics Subject Classification. 65N3, 65N55. Received: September 3,. Revised: October,.. Introduction In any domain decomposition method, one has to define how the continuity between the subdomains is to be enforced. Different approaches have been proposed: Iterative procedures, enforcing that the approximate solution or its normal derivative or combinations thereof should be continuous across interfaces. This forms the basis for the standard Schwarz alternating method as defined, e.g., by Lions [6]. Direct procedures, using Lagrange multiplier techniques to achieve continuity. Different variants have been proposed, e.g., by Le Tallec and Sassi [5], and Bernadi et al. [8]. The multiplier method has the advantage of directly yielding a solvable global system. However, in the latter method, new unknowns (the multipliers) must be introduced and solved for. The method must then either satisfy the inf-sup condition, which necessitates special choices of multiplier spaces (such as mortar elements, cf. [8]), or then stabilization techniques (cf. [3,4,9]) must be used. In this paper, we consider a third possibility, i.e. Nitsche s method [7], which was originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary conditions in the definition of the finite element spaces. This method has later been used by Arnold [] for the discretization of second order elliptic equations by discontinuous finite elements. In earlier papers [8, 9] we have pointed out the close connection between Nitsche s method and stabilized methods and proposed it as a mortaring method. In this paper we will give a more detailed analysis of this domain decomposition technique where independent approximations Keywords and phrases. Nitsche s method, domain decomposition, non-matching grids. Institute of Applied Mathematics, University of Heidelberg, INF 94, 69 Heidelberg, Germany. Department of Applied Mechanics, Chalmers University of Technology, 4 96 Göteborg, Sweden. e-mail: hansbo@am.chalmers.se 3 Institute of Mathematics, Box, 5 Helsinki University of Technology, Finland. c DP Sciences, SMAI 3
R. BCKR T AL. are used on the different subdomains. The continuity of the solution across interfaces is enforced weakly, but in such a way that the resulting discrete scheme is consistent with the original partial differential equation. Under some regularity assumptions we derive both aprioriand a posteriori error estimates. We also give numerical results obtained with the method. Although we discuss its application to domain decomposition, the same technique is also suited for other applications, e.g., to handle diffusion terms in the discontinuous Galerkin method [, 6]; to simplify mesh generation (different parts can be meshed independently from each other); finite element methods with different polynomial degree on adjacent elements; new finite element methods such as linear approximations on quadrilaterals.. The domain decomposition method In this section we will introduce the mortaring method based on the classical method of Nitsche. We will perform a classical stability and apriorierror analysis. For simplicity, we consider the model Poisson problem, i.e. of solving the partial differential equation u = f in Ω, u = on Ω, () where Ω is a bounded domain in two or three space dimensions and f L (Ω). Likewise for ease of presentation, we consider only the case where Ω is divided into two non-overlapping subdomains Ω and Ω,Ω Ω, with interface = Ω Ω. We further assume that the subdomains are polyhedral (or polygonal in R ) and that is polygonal (or a broken line). This equation can be written in weak form as: find u H (Ω) such that where a(u, v) = a(u, v) =(f,v), v H (Ω), () Ω u vdx, (f,v) = Ω fvdx, (3) and H (Ω) is the space of square-integrable functions, with square-integrable first derivatives, that vanish on the boundary Ω of Ω. Our discrete method for the approximate solution of () is a nonconforming finite element method which is continuous within each Ω i and discontinuous across. We start by rewriting the original problem () as two equations and the interface conditions: u i = f in Ω i, i =,, u i = on Ω Ω i, i =,, u u = on, (4) u + u = on. Here, n i is the outward unit normal to Ω i. We will perform our analysis under the following regularity assumption. Assumption.. The solution of () satisfies u H s (Ω), withs>3/.
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS With this assumption it holds u i / i L () and the two problems () and (4) are equivalent (see for example []) with: u Ωi = u i, i =,. (5) In the following we will therefore write u =(u,u ) V V with the continuous spaces V i = { v i H (Ω i ): v i / i L (), v i Ω Ωi = }, i =,. To formulate our method, we suppose that we have regular finite element partitionings Th i Ω i into shape regular simplexes. These two meshes induce trace meshes on the interface of the subdomains G i h = { : = K, K T i h} (6) By h K and h we denote the diameter of element K Th i and Gi h, respectively. For the purpose of the apriorianalysis, we also define In our analysis we will need the following assumption. h =max{h K,h : K T i h, G i h,, } Assumption.. There exists positive constants C,C such that C h h C h holds for all pairs (, ) Th T h,with. We seek the approximation u h =(u,h,u,h )inthespacev h = V h V h,where V h i = { v i V i : v i K is a polynomial of degree p for all K T i h}, choosing for simplicity p constant on all subdomains. On the interface we will use the notation for the jump, for the average and v, w S := S [v]] := v v (7) {v} := v + v (8) vwds, for S Gh i, or S =. (9) Finally, we denote n := n = n. () With this notation we have { } v = v v () and hence { } u = u = u = u () The methods of Nitsche [7] and Arnold [] now give the following domain decomposition method. Mortar method. Find u h V h such that a h (u h,v)=f h (v) v V h, (3)
R. BCKR T AL. with a h (w, v) := [( w i, v i ) Ωi + γ { w }, [v]] G i h { v [w]], [v ] ] }, [w]] (4) and f h (v) := where γ> is chosen sufficiently large, see Lemma.6 below. The first observation is that this formulation gives a consistent method. Lemma.3. The solution u =(u,u ) to (4) satisfies (f,v i ) Ωi, (5) a h (u, v) =f h (v) v V. (6) Proof. Multiplying the first equation in (4) with v i, integrating over Ω i, using Greens formula and the relations () yields f h (v) = = (f,v i ) Ωi = ( u i, v i ) Ωi [ ] ui ( u i, v i ) Ωi,v i i { } u, [v ] (7) Since [u] = on we have = { } v, [u] + γ Gh i [[u], [v]]. (8) Adding (7) and (8) the gives the claim. For the stability analysis below we need the following mesh-dependent dual norms. To emphasize that is to be considered as a part of Ω i we write i. v /,h, i = v L () (9) G i h and v /,h, i = h v L (), () which satisfy The analysis will be performed using the norms G i h v, w v /,h,i w /,h,i. () ( ) v,h = v i L(Ωi) + [v]] /,h,i ()
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS 3 and v,h = v,h + v i i /,h, i (3) The following estimate is readily proved by local scaling. Lemma.4. There is a positive constant C I such that v i i C I v i L (Ω i) /,h, i v i Vi h. (4) For linear elements v is constant on each element and then it is particularly easy to give a bound for the constant C I, see Remark. below. A immediate consequence of the above lemma is the equivalence of norms in the finite element subspace. Lemma.5. The norms,h and,h are equivalent on the subspace V h. The stability of the method can now be proved. Lemma.6. Suppose that γ>c I /4. Then it holds a h (v, v) C v,h v V h. Proof. From the definition (4), the relation (), and the preceding lemmas we get a h (v, v) = = [ ] { } v i L(Ωi) v + γ [v [v]], ] /,h,i [ ] v i L(Ωi) + γ [v [v ], v + [v]], v ] /,h,i [ v i L + γ [[v (Ω i) ] /,h, i [v]], v ] i i [ ] v i L + γ v i (Ω i) [v]] /,h, i [v ] /,h,i i /,h,i [( C ) ( I v i L(Ωi) ε + γ ε ) ] [v]] /,h,i C v,h C v,h, for γ>ε/and choosing ε>c I /. The following interpolation estimates holds, cf. []. Lemma.7. Suppose that u H s (Ω), with3/ <s p +. Then it holds inf v V h u v,h Ch s u Hs (Ω) (5) and inf v V h u v,h Ch s u H s (Ω). (6)
4 R. BCKR T AL. We are now able to prove the following apriorierror estimate. Theorem.8. Suppose γ>c I /4. Then it holds If u H s (Ω), for3/ <s p +,then u u h,h C inf v V h u v,h. u u h,h Ch s u Hs (Ω). Proof. Since the bilinear form a h is bounded with respect to the norm,h, the first inequality follows from the stability estimate of Lemma.6 and the triangle inequality. The second estimate then follows from Lemma.7. Remark.9. As [u] = on the error estimate gives [u h ]] /,h,i = O(h s ). For quasi-uniform meshes it then holds [u h ]] = L() O(hs / ). Remark.. The presented method resembles a mesh-dependent penalty method, but with added consistency terms involving normal derivatives across the interface. Note that the formulation allows us to deduce optimal order error estimates with preserved condition number of O(h ) for a quasiuniform mesh. Pure penalty methods, in contrast, are not consistent, and optimal error estimates require degrading the condition number for higher polynomial approximation (cf. [5]). Remark.. The form a h (, ) in (4) is symmetric and positive definite, which is natural as the problem to be approximated has the same properties. With this there exists fast solvers, such as preconditioned conjugate gradients, for the resulting matrix problem. If the bilinear form is changed to a h (w, v) := [( w i, v i ) Ωi + γ { w }, [v]] + G i h { v [w]], [v ] ] }, [w]], (7) then one obtains a method which is stable for all positive values of γ: a h (v, v) C v,h v V h, γ >. If the Laplace operator is a part of a problem which is not symmetric, then it might be practical to use this nonsymmetric bilinear form. See [, ] for applications to convection diffusion problems. Remark.. For linear elements v i (v i Vi h ) is constant on each element and hence for K Ti h base Gh i we have v i i =meas() v i, i and v i,k v i i =meas(k) v i i and hence it holds v i h i,,k h meas() meas(k) with the (8) (9) v i,k. (3)
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS 5 Hence, once the shape regularity of the mesh is specified, one has a bound for C I. In a practical implementation easier would be to replace the weight γ in (4) with a parameter α K = α meas() meas(k), (3) with α>/4 fixed. With this all results of the paper hold. Remark.3. We have performed our analysis assuming that the solution is sufficiently smooth, i.e. that u H s (Ω) with s>3/. The analysis can be extended for less regular solutions if the corner and transmission singularities are carefully taken into account, cf. the recent papers [,3]. 3. A POSTRIORI error estimates We will first consider control of the error e = u u h in the mesh dependent energy norm,h. We define the local and global estimators as [[ ]] K (u h ) = h K f + u h L + h uh (K) K K L ( K) + K [u h ] L ( K) (3) and [ ] / (u h )= K (u h ). (33) To be able to control the normal derivatives across the interface, we are forced to introduce a saturation assumption. This assumption is consistent with the Assumption. on the regularity of the solution and the interpolation estimates of Theorem.7. Assumption 3.. There is a constant C such that u u h,h C u u h,h. (34) We also remark that an analog assumption is used in the context of a posteriori error estimates for the mortar element method, see Wohlmuth []. The a posteriori estimate is now the following. Theorem 3.. Suppose that the Assumption 3. is valid. Then there is a positive constant C such that Proof. We denote e = u u h. By Lemma.3 we have { } e,h = a h(e, e)+ [e], +( γ) u u h,h C(u h ). (35) Gh i [[e], [e] := R + R, (36) with and R := a h (e, e), { } R := [e], +( γ) [e] /,h, i.
6 R. BCKR T AL. Since [e] =[u h ] Assumption 3. gives R = [u h ]], [u h ], +( γ) [e] /,h, i ( ) i C [[u h ] /,h,i i + [u h ] /,h,i [e] /,h,i /,h,i C e,h ( [u h ] /,h,i ) C e,h (u h ). (37) Next, let π h e be the Clément interpolant to e, which satisfies ( ( h K e π he L + (K) h K e π he L ( K) ) ) / C e,h (38) and e π h e,h C e,h. (39) From the consistency (.3) we have a h (e, π h e)=. Hence R = a h (e, e) =a h (e, e π h e) = ( e i, (e i π h e i )) Ωi + γ Integrating by parts on each K T h yields G i h [e], [e π h e] { } { } (e πh e), [e π h e], [e] := S + S + S 3 + S 4. (4) S = ( e i, (e i π h e i )) Ωi (4) = := T + T. ( ) ( e, e π h e) K +,e π h e K K Since on K T h it holds e = u + u h = f + u h, the first term above is estimated using (38) T = ( e, e π h e) K (4) ( ) / ( ) / h K u h + f L (K) h K e π he L (K) ( ) / h K u h + f L (K) e,h.
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS 7 Let I h be the collection of element sides in the interiors of the subdomains Ω i. The boundary integrals in (4) above we now split into those in I h and those lying on the interface : T =,e π h e (43) K = I h,e π h e K +,e π h e +,e π h e The integrals over the interior sides can now be grouped together two by two yielding the estimate [ [[ ]] ],e π h e I h C h K uh / [ / K e π he L K ( K)] (44) L ( K) [ [[ ]] ] C uh / e,h, K where (38) was used. Using the definition () we have { S 3 = = h K }, [e π h e] =,e π h e L ( K) +,e π h e (e π h e ) +, (e π h e ) (45) Combining these terms with two last terms in (43) gives { },e π h e +,e π h e, [[e π h e] ( = ) + (,e π h e + + ), (e π h e ) = +,e π h e + +,e π h e = (e e ),e π h e + (e e ),e π h e = [[ ]],e π h e + [[ ]],e π h e = [[ ]] uh,e π h e + [[ ]] uh,e π h e [ ] / (u h) K e π he L ( K) C(u h ) e,h. (46) From (4) to (46) we now get S + S 3 C e,h (u h ). (47)
8 R. BCKR T AL. From the Clément estimate (38) have S = γ Gh i [e], [e π h e] C e,h (u h ). To estimate the last term S 4 we use Lemma.4 and the Assumption 3. { } { } { } (e πh e) πh e S 4 =, [e] =, [[e] +, [e] C e,h (u h ). (48) Hence, collecting the estimates (4) to (48) yields which together with (37) proves the estimate R C e,h (u h ), (49) u u h,h = e,h C(u h ). (5) The claim then follows from (34). Next, we consider the error in the L -norm. This is measured with the estimator L(u h )= [ h K K (u h ) ] /. (5) For the a posteriori estimate we as usual, need the H -regularity but not Assumption 3.. Theorem 3.3. Let z be the solution to the problem z = g in Ω, z = on Ω, (5) and suppose that the shift theorem is valid. Then there is a positive constant C such that z H (Ω) C g L (Ω) (53) u u h L(Ω) CL(u h). (54) Proof. We again write z =(z,z ) and note that z + z = and { } z = z = z
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS 9 Choosing g = u u h in (5) we then get u u h L (Ω) = = = [ zi ( z i, (u i u i,h )) Ωi,u i u i,h i [ ] zi ( z i, (u i u i,h )) Ωi +,u i,h i ( z i, (u i u i,h )) Ωi + { z }, [[u h ] i i ] (55) We let π h z V h be the Clément interpolant to z satisfying ( ( h 4 K z π hz L (K) + h 3 K z π hz L ( K) + h K (z π h z) K L ( K) )) / C z H(Ω). (56) From the consistency condition we have a h (u u h,π h z) =, and since [z ] =, [u] =, on the interface, we get =a h (u u h,π h z) (57) = ( (u i u i,h ), π h z i ) Ωi + γ [u u h ]], [π h z ] G i h { } { (u uh ) πh z, [π h z ] = ( (u i u i,h ), π h z i ) Ωi + γ { (u uh ) + }, [z π h z ] + G i h }, [u u h ] { πh z [ u h ], [z π h z ] }, [u h ]] Substracting (57) from (55) gives u u h L = (Ω) ( (z i π h z, (u i u i,h )) Ωi γ G i h [[ u h ], [z π h z ] { } { } (u uh ) (z πh z), [z π h z ] +, [u h ]] := R + R + R 3 + R 4. (58)
R. BCKR T AL. Using Schwarz inequality and the Clément estimate (56) we get R + R 4 C ( ( h 3 K z π hz L ( K) + h K C z H (Ω) L(u h) C u u h L(Ω) L(u h). (z π h z) K L ( K) )) / h [[u h ] L() G i h / Integrating by parts yields R + R 3 = [ ( (z i π h z, (u i u i,h )) Ωi { } (u uh ), [z π h z ] = { (z π h z, (u h u)) K (u uh ) + K { (u uh ) =,z π h z K } }, [z π h z ] (z π h z, u h + f) K + (u uh ),z π h z I h (ui u i,h ) +,z i π h z i i { } (u uh ), [z π h z ] The boundary terms in the second term above are grouped together two by two yielding jump terms in the normal derivative. This gives the estimate (u uh ) (z π h z, u h + f) K + I h ( ( C h 4 K,z π h z z π hz L + (K) h 3 K z π hz L ( K) ( [[ ]] h 4 K f + u h L + (K) h3 uh K K C z H (Ω) L(u h) C u u h L(u L(Ω) h). L ( K) ) ) / ) / (59)
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS The remaining two terms are estimated as follows { } (ui u i,h ) (u uh ),z i π h z i, [z π h z ] i ui ui,h u =,z i π h z i,z i π h z i i i, [z π h z ] + { } ui,h uh =,z i π h z i +, [[z π h z ] i u,h u,h =,z π h z,z π h z + u,h,z π h z u,h,z π h z + u,h,z π h z = u,h,z π h z u,h,z π h z u,h,z π h z = u,h + u,h,z π h z u,h + u,h,z π h z = [[ ]] uh,z π h z [[ ]] uh,z π h z ( ) / CL(u h ) z π hz L ( K) h 3 K CL(u h ) z H (Ω) CL(u h) u u h L(Ω). Combining (59) and (6) gives { } uh, [z π h z ] u,h,z π h z u,h,z π h z (6) R + R 3 CL(u h ) u u h L(Ω) which together with (59) and (58) proves the claim. 4. Numerical examples 4.. Numerical verification of the aprioriestimates To verify the aprioriestimates, we choose the model problem of a unit square with exact solution u = xy( x)( y) corresponding to a right-hand side of f =(x x + y y ). The domain is divided by a vertical slit at x =.7. Two different triangulations were used: one matching and one non-matching (see Fig. ). In Figure (left-hand side) we give the convergence in the broken energy norm. The dashed line is the non-matching grid computation. Both meshes show the same convergence with slope.95. which is close to the theoretical value of. On the right-hand side we show the convergence of the L -norm of the jump term (dashed line for the non-matching grid). Here we obtain a better convergence (slope.5) for the matching grids than for the non-matching grids (slope.57, close to the theoretical value of 3/). 4.. Adaptive computations We present results of adaptive computations on the L-shaped domain Ω=(, ) (, ) \ (/, ) (, /).
R. BCKR T AL. Figure. Matching and non-matching grids. log(energy error) -3.6-3.8-4 -4. -4.4-4.6 log(interface jump) -8-8.5-9 -9.5 - -.8 -.6 -.4 -. - -.8 -.6 -.4 log(h) -.5-3.5-3 -.5 - -.5 - -.5 log(h) Figure. Convergence in energy and convergence of the jump term. The problem is boundary driven (f = ), with boundary data corresponding to the exact solution u = r /3 sin (θ/3) in polar coordinates (with origin at (/, /)). We let Ω = (, /) (, ) and Ω = (/, ) (/, ), and use a non-matching triangulation. The purpose of this example is not to obtain exact error control, but rather to show how the adaptive algorithm behaves with respect to the elements adjacent to the interface. We consider adaptive control of the L -norm error, but we have not made any attempt to measure the constant in the inequality (53); instead we have simply tuned the interpolation constants to approximately match the exact error. In Figures 3 and 4 we show the first and last (adapted) meshes resulting from equilibrating the error distribution over the set of elements (for details, see [7,4]). In Figure 5 we show the exact and estimated L -errors on the sequence of meshes, which show a reasonable agreement. For more exact error control, more computational effort must be invested.
A FINIT LMNT MTHOD FOR DOMAIN DCOMPOSITION WITH NON-MATCHING GRIDS 3 Figure 3. First mesh and final adapted mesh..8.7.6.5.4.3....4.6.8..4.6.8 Figure 4. levation of the solution. 4.3. Difference between matching and non-matching meshes In the previous example, the adaptive algorithm produced a slightly finer mesh on the interface. One natural question is then whether the effect of non-matching actually does lead to larger errors. In Figures 6 and 7 we show two different meshes for solving the problem described in Section 4., and the corresponding nodal interpolants of the errors. The interface is situated at x =/, and it is noted that the error on the interface is markedly larger than in the interior of the domains only in the case of non-matching meshes.
4 R. BCKR T AL.. xact L-error stimated L-error..8.6.4. 3 4 5 6 7 8 9 Figure 5. stimated and exact errors in the L (Ω)-norm..9.8 x -4.7 5 4.6 3.5.4.3.. -.8.6.4.4.6.8..4.6.8.. Figure 6. Matching meshes and nodal errors. x -4.9.8.7 8 6.6 4.5.4 -.3-4....4.6.8-6 -8..4.6.8..4.6.8 Figure 7. Non-matching meshes and nodal errors.
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