Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries

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1 Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries Laurence Halpern, Caroline Japhet, and Jérémie Szeftel 3 LAGA, Université Paris XIII, Villetaneuse 9343, France, halpern@math.univ-paris3.fr LAGA, Université Paris XIII, Villetaneuse 9343, France; CSCAMM, University of Maryland College Park, MD 74 USA, japhet@cscamm.umd.edu, the first two authors are partially supported by french ANR COMMA and GdR MoMaS. 3 Département de mathématiques et applications, Ecole Normale supérieure, 45 rue d Ulm, 753 Paris cedex 5 France, Jeremie.Szeftel@ens.fr Introduction In many fields of applications it is necessary to couple models with very different spatial and time scales and complex geometries. Amongst them are ocean-atmosphere coupling and far field simulations of underground nuclear waste disposal. For such problems with long time computations, a splitting of the time interval into windows is essential. This allows for robust and fast solvers in each time window, with the possibility of nonconforming space-time grids, general geometries, and ultimately adaptive solvers. Optimized Schwarz Waveform Relaxation OSWR methods were introduced and analyzed for linear advection-reaction-diffusion problems with constant coefficients in [, 3, 9]. All these methods rely on an algorithm that computes independently in each subdomain over the whole time interval, exchanging space-time boundary data through optimized transmission operators. They can apply to different space-time discretization in subdomains, possibly nonconforming and need a very small number of iterations to converge. Numerical evidences of the performance of the method with variable smooth coefficients were given in [9]. An extension to discontinuous coefficients was introduced in [ 4], with asymptotically optimized Robin transmission conditions in some particular cases. In [, 6], semi-discretization in time in one dimension was performed using discontinuous Galerkin, see [8, ]. In [7], we extended the analysis to the two dimensional case. We obtained convergence results and error estimates for rectangular or stripsubdomains. For the space discretization, we extended numerically the nonconforming approach in [5] to advection-diffusion problems and optimized order transmission

2 76 L. Halpern et al. conditions, to allow for non-matching grids in time and space on the boundary. The space-time projections between subdomains were computed with an optimal projection algorithm without any additional grid, as in [ 5]. In [7], two dimensional simulations with continuous coefficients were presented. We present here new results in two directions: we extend the proof of convergence of the OSWR algorithm to nonoverlapping subdomains with curved interfaces. We also present simulations for two subdomains, with piecewise smooth coefficients and a curved interface, for which no error estimates are available. We finally present an application to the porous media equation. We consider the advection-diffusion-reaction equation, t u + bu ν u+cu = f in R N,T, with initial condition u, and N =. The advection, diffusion and reaction coefficients b, ν and c, are piecewise smooth, we suppose ν ν > a.e.. The Continuous OSWR Algorithm We consider a decomposition into nonoverlapping subdomains Ω i,i {,..., I}, organized as depicted in Fig.. The interfaces between the subdomains are supposed to be flat at infinity. For any i {,..., I}, Ω i is the boundary of Ω i, n i the unit exterior normal vector to Ω i, N i is the set of indices of the neighbors of Ω i.for j N i, Γ i,j is the common interface. Ω i Ω i Fig.. Decomposition in subdomains. Left: Robin transmission conditions, right: second order transmission conditions. Following [,, 3, 4], we introduce the boundary operators S i,j acting on functions defined on Γ i,j : S i,j ϕ = p i,j ϕ + q i,j t ϕ + Γi,j r i,j ϕ s i,j Γi,j ϕ, with respectively Γ and Γ the gradient and divergence operators on Γ. p i,j,q i,j, r i,j, s i,j are real parameters. q i,j =, will be referred to as a Robin operator. We introduce the coupled problems t u i + b i u i ν i u i +c i u i = f in Ω i,t νi ni b i n i ui + S i,j u i = νj ni b j n i uj + S i,j u j on Γ i,j,t,j N i.

3 Space-Time Nonconforming OSWR for Heterogeneous Problems 77 As coefficients ν and b are possibly discontinuous on the interface, we note, for s Γ i,j, ν i s = lim ε νs εn i. The same notation holds for b. Under regularity assumptions, solving is equivalent to solving for i {,..., I} with u i the restriction of u to Ω i. We now introduce an algorithm to solve. An initial guess g i,j is given in L,T Γ i,j for i {,..., I},j N i. We solve iteratively t u k i + b i u k i ν i u k i +c iu k i = f in Ω i,t, νi ni b i n i u k i + S i,j u k i = νj ni b j n i u k j + S i,j u k j on Γ i,j,t,j N i. 3 with the convention ν i ni b i n i u i + S i,j u i = g i,j, j N i. Theorem. Assume b i W, Ω i N, ν i W, Ω i, p i,j W, Γ i,j with p i,j > a.e.. Ifq i,j =,orifq i,j = q > with r i,j W, Γ i,j N, r i,j = r j,i on Γ i,j, s i,j W, Γ i,j, s i,j >, s i,j = s j,i on Γ i,j, the algorithm 3 converges in each subdomain to the solution of problem. Proof. We first need some results in differential geometry. For every j N i, the normal vector n i can be extended in a neighbourhood of Γ i,j as a smooth function ñn i with length one. Let ψ i,j C Ω i, such that ψ i,j in a neighbourhood of Γ i,j, ψ i,j in a neighbourhood of Γ i,k for k N i,k j and j N i ψ i,j > on Ω i. Let ñn i be defined on a neighbourhood of the support of ψ i,j. We can extend the tangential gradient and divergence operators in the support of ψ i,j by: Γi,j ϕ := ϕ ñn i ϕñn i, Γi,j ϕ := ϕ ϕ ñn i ñn i. It is easy to see that Γi,j ϕ Γi,j = Γi,j ϕ, Γi,j ϕ Γi,j = Γi,j ϕ and for ϕ and χ with support in suppψ i,j,wehave Γi,j ϕ χdx= Ω i ϕ Γi,j χdx. Ω i 4 Now we prove Theorem. The key point is to obtain energy estimates for the homogeneous problem, i.e. for f = u =. We sketch the proof in the most difficult case q i,j = q>. For the geometry, we consider the case depicted in the right part of Fig.. In that case Ω i has at most two neighbours with We set ϕ i = ϕ L Ω i, ϕ i = ν i ϕ L Ω, ϕ i i, = ϕ L Ω i, ϕ i,, = ϕ W, Ω i and β i = pi,j+p j N i ψ i,j β i,j with β i,j = j,i.. We multiply the first equation of 3byβ i uk i, integrate on Ω i,t then integrate by parts in space,

4 78 L. Halpern et al. t β iu k i t i + β i u k i τ, i dτ β i b i β i u k i dx dτ Ω i + c i + Ω i b i βi u k i dx dτ ν i β i u k i dx dτ Ω i βi,j ν i ni u k i b i n i u k i Γ i,j uk i dσ dτ =. 5. We multiply the first equation of 3by t u k i, integrate on Ω i,t and integrate by parts in space, t u k i i dτ + uk i t i + c i u k i + b i u k i t u k i dx dτ Ω i ν i ni u k i t u k i dσ dτ =. 6 Γ i,j 3. We multiply the first equation of 3by Γi,j ψi,j r i,j u k i integrate on Ω i,t integrate by parts in space to obtain t u k i Γi,j ψi,jr i,j u k i dx dτ+ b i u k i Γi,j ψi,jr i,j u k i dx dτ Ω i Ω i t + c i u k i Γi,j ψi,jr i,j u k i dx dτ ν i ni u k i Γi,j r i,j u k i dσ dτ Ω i Γ i,j t 4 Γi,j u k i i dτ C ν i u k i i + β iu k i i dτ We multiply the first equation of 3 by Γi,j ψi,j s i,j Γi,j u k i integrate on Ω i,t, integrate by parts in space using 4. Using that ν i u k i Γi,j ψi,j s i,j Γi,j u k i dx dτ Ω i t Γi,j,u k i i dτ C ν i u k i i dτ, we obtain ψ i,j si,j Γi,j u k i t i + Γi,j u k i i dτ t + ψi,j s i,j c i Γi,j u k i dx dτ + ν i ni u k i Γ i,j s i,j Γi,j u k i dσ dτ Ω i Γ i,j t b i u k i Γi,j ψi,j s i,j Γi,j u k i dx dτ + C ν i u k i i dτ. 8 Ω i We add 6, 7 and 8, multiply the result by q, and add it to 5. We use ab a ε + ε b in the integral terms in the right-hand side, simplify with the left-hand side, and obtain

5 Space-Time Nonconforming OSWR for Heterogeneous Problems 79 β i u k i t i + q uk i t i +q ψ i,j si,j Γi,j u k i t i + β i u k i τ, i dτ + q t u k i i dτ + q 8 Γi,j u k i i dτ t q ν i ni u k i t u k i + Γ i,j r i,j u k i Γ i,j s i,j Γi,j u k i dσ dτ Γ i,j t βi,j ν i ni u k i b i n i u k i Γ i,j uk i dσ dτ q b i i,, + c i i, u k i t i + C β i u k i i dτ + q ν i u k i i dτ. 9 Recalling that s i,j = s j,i on Γ i,j and r i,j = r j,i on Γ i,j, we use now the identity: νi ni u k i b i n i u k i + S i,j u k i νi ni u k i b i n i u k i S j,i u k i =4 βi,j ν i ni u k i b i n i u k i uk i + qν i ni u k i tu k i + Γ i,j r i,j u k i 4 Γi,j s i,j Γi,j u k i +qp i,j p j,i b i n i t u k i + Γi,j r i,j u k i Γi,j s i,j Γi,j u k i u k i +p i,j + p j,i p i,j p j,i b i n i u k i. Replacing into 9, we obtain β i u k i t i + q u k i t i +q ψ i,j si,j Γi,j u k i t i + β i u k i τ, i dτ + q t u k i i dτ + νi ni u k i 4 b i n i u k i S j,iu k i dσ dτ Γ i,j + q 8 Γi,j u k i i dτ νi ni u k i b i n i u k i + S i,j u k i dσ dτ 4 Γ i,j t + p i,j +p j,i p i,j +p j,i +b i n i u k i dσ dτ+ q Γ i,j b i i,, + c i i, u k i t i + q p i,j +p j,i +b i n i t u k i + Γi,j r i,j u k i Γi,j s i,j Γi,j u k i u k i dσ dτ Γ i,j + C β i u k i i dτ + q ν i u k i i dτ. In order to estimate the fourth term in the right-hand side of, we observe that p i,j +p j,i +b i n i u k i tu k i dσ dτ = p i,j +p j,i +b i n i u k i Γ i,j t dσ. Γ i,j By the trace theorem in the right-hand side, we write: p i,j + p j,i +b i n i u k i t u k i dσ dτ C u k i t i ν i u k i t i, Γ i,j

6 8 L. Halpern et al. and t u k i t i = t u k i u k i t u k i i Ω i u k i i, we obtain q t p i,j + p j,i +b i n i u k i t u k i dσ dτ Γ i,j q t u k i i dτ + q 8 4 uk i t i +C β i u k i i dτ. 3 Moreover, integrating by parts and using the trace theorem, we have: q Γi,j s i,j Γi,j u k i i,j + p j,i +b i n i u k i dσ dτ Γ i,j q 6 Γi,j u k i i dτ + C Γi,j u k i i dτ + β i u k i i dτ. 4 Using, we estimate the third term in the right-hand side of by: q b i i,, + c i i, u k i t i q t u k i i dτ + C β i u k i i dτ. 5 8 Replacing 4, 3 and 5 in, then using the transmission conditions, we have: β i u k i t i + q uk i t i +q ψ i,j si,j Γi,j u k i t i + β i u k i τ, i dτ + q t u k i 4 i dτ + q 6 Γi,j u k i i dτ + νi ni u k i b i n i u k i S j,i u k i dσ dτ 4 Γ i,j νj ni u k j b j n i u k j + S i,j u k j dσ dτ 4 Γ i,j + C β i u k i i dτ + q ν i u k i i dτ. We now sum up over the interfaces j N i, then over the subdomains for i I, and over the iterations for k K, the boundary terms cancel out, and with αt = 4 i {,...,I} j N νj i Γ i,j ni u i b j n i u j + S i,juj dσ dτ, we obtain for any t,t, β i u k i t i + q ν i u k i t i + ν β i u k i i dτ k {,...,K} i {,...,I} αt+c β i u k i i dτ + q ν i u k i i dτ. k {,...,K} i {,...,I}

7 Space-Time Nonconforming OSWR for Heterogeneous Problems 8 We conclude with Gronwall s lemma that the sequence converges in L,T; H Ω i. 3 Numerical Results We recall the discrete time nonconforming Schwarz waveform relaxation method developed in [7]. Let T i be the time partition in subdomain Ω i, with N i +intervals In, i and time step kn i. We define interpolation operators I i and projection operators P i in each subdomain as in [7], and we solve t I i Ui k+ bu i k ν i Ui k+c i Ui k = P i f in Ω i,t, νi ni b i n i U k i + S i,j Ui k = P i ν j ni b j n i U k j + S i,j Uj k on Γi,j,T, with S i,j U = p i,j U + q i,j t I i U+ Γi,j r i,j U s i,j Γi,j U, and S i,j U = p i,j U + q i,j t I j U+ Γi,j r i,j U s i,j Γi,j U. The coefficients p i,j and q i,j are defined through an optimization procedure, see [], restricted to values such that the subdomain problems are well-posed. The time semi-discrete analysis was performed in [7] in the case b =. For the space discretization, we use the nonconforming approach in [ 5] extended to problem and order transmission conditions, to allow non-matching grids in time and space on the boundary. We have implemented the algorithm with P finite elements in space in each subdomain. Time windows are used in order to reduce the number of iterations of the algorithm. To reduce the number of parameters and following [ ], we choose r i,j = Π Γi,j b j with Π Γi,j the tangential trace on Γ i,j, and s i,j = ν j even though the present analysis does not cover this case. The optimized parameters are constant along the interface. They correspond to a mean value of the parameters obtained by a numerical optimization of the convergence factor. We first give an example of a multidomain solution with one time window. The physical domain is Ω =,,, the final time is T =. The initial value is u =.5e x.55 +y.7 and the right-hand side is f =. The domain Ω is split into two subdomains Ω =,.5, and Ω =.5,,. The reaction factor c is zero, the advection and diffusion coefficients are bb =,, ν =. y, and b =.,, ν =.sinxy. The mesh size over the interface and time step in Ω are h =/3 and k =/8, while in Ω, h = /4 and k = /94. On Fig., we observe, at final time T =, a very good behavior of the multidomain solution after 5 iterations. The relative error with the one domain solution is of the same order as the error of the scheme. We analyze now the precision in time. The space mesh is conforming and the converged solution is such that the residual is smaller than 8. We compute a variational reference solution on a time grid with 4,96 time steps. The nonconforming solutions are interpolated on the previous grid to compute the error. We start with a time grid with 8 time steps for the left domain and 94 time steps for the right domain. Thereafter the time steps are divided by several times. Figure 3 left shows

8 8 L. Halpern et al. y x Fig.. Nonconforming DG-OSWR solution after 5 iterations. conforming finner grid conforming coarser grid nonconforming domain nonconforming domain k Order Robin Error 3 4 Error Number of refinements 5 5 Iterations Fig. 3. Error between variational and DG-OSWR solutions versus the refinement in time left, and versus the iterations right. the norms of the error in L I; L Ω i versus the number of refinements, for both subdomains. First we observe the order in time for the nonconforming case. This fits the theoretical estimates, even though we have theoretical results only for Robin transmission conditions. Moreover, the error obtained in the nonconforming case, in the subdomain where the grid is finer, is nearly the same as the error obtained in the conforming finer case.

9 Space-Time Nonconforming OSWR for Heterogeneous Problems 83 The computations are done using Order transmissions. Indeed, the error between the multidomain and variational solutions decreases much faster with the Order transmissions conditions than with the Robin transmissions conditions as we can see on Fig. 3 right, in the conforming case. We now consider advection-diffusion equations with discontinuous porosity: ω t u + bu ν u =. The physical domain is Ω =,,, the final time is T =.5. The initial value is u =.5e x.7 +y.5. Domain Ω is split into two subdomains Ω,.5 and Ω,.5 with sinπs 8, s, <s<a parametrization of the interface, as in Fig. 4. The advection and diffusion coeffi y x Fig. 4. Domain decomposition with Ω left and Ω right. cients are b = sin π y cosπx, 3cos π y sinπx, ν =.3, ω =., and b = b, ν =., ω =. We consider first a conforming grid in space. The mesh size over the interface is h =/4 and time step in Ω is k =/8, while in Ω, k =/94. On Fig. 5, we observe, at final time T =.5, that the approximate solution computed using ten time windows and 3 iterations in each time window is close to the variational solution computed in one time window on the conforming finer space-time grid as shown on the error. We now consider nonconforming grids in space as shown on Fig. 4. The mesh size over the interface and time step in Ω are h =/4 and k =/8, while in Ω, h =/8 and k =/94. On Fig. 6, we observe, at final time T =.5, that the approximate solution computed using 5 iterations in one time window is close to the variational solution computed on the conforming finer space-time grid. On Fig. 7 we observe the precision versus the mesh size and time step. The converged solution is such that the residual is smaller than 8. A variational reference solution is

10 84 L. Halpern et al. computed on a time grid with,48 time steps and 384 mesh grid. The space-time nonconforming solutions are interpolated on the previous grid to compute the error. We start with a time grid with 3 time steps and 4 mesh size for the left domain and time steps and mesh size for the right domain and divide by the time step and mesh size several times. Figure 7 shows the norms of the error in L I; L Ωi versus the time steps, for both subdomains. We observe the order for the nonconforming space-time case, even though we have theoretical results only for the time semi-discretized case in [7]. 4 x y x Fig. 5. Error between variational and DG-OSWR solutions, at final time, after time windows and 3 iterations per window. 4 Conclusions We have analyzed the continuous algorithm for variable discontinuous coefficients and general decompositions. We have shown numerically that the method preserves the order of the one domain scheme in the case of discontinuous variable coefficients, nonconforming grids in space and time and a curved interface. An analysis of the influence of the decomposition in time windows is in progress.

11 Space-Time Nonconforming OSWR for Heterogeneous Problems x Fig. 6. DG-OSWR solution at final time, after 5 iterations. Error versus the time step Domain Domain Slope Error y Time step Fig. 7. Error curves versus the time step.

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