Caroline Japhet

The optimized order 2 method: Application to convection–diffusion problems

We present an iterative, non-overlapping domain decomposition method for solving the convection–diffusion equation. A reformulation of the problem leads to an equivalent problem, where the unknowns are on the boundary of the subdomains [F. Nataf, F. Rogier, E. de Sturler, in: A. Sequeira (Ed.), Navier–Stokes Equations on Related Nonlinear Analysis, Plenum Press, New York, 1995, pp. 307–377]. The solving of this interface problem by a Krylov type algorithm [Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996] is done by the solving of independent problems in each subdomain, so it permits to use efficiently parallel computation. In order to have very fast convergence, we use differential interface conditions of order 1 in the normal direction and of order 2 in the tangential direction to the interface, which are optimized approximations of absorbing boundary conditions [C. Japhet, Methode de decomposition de domaine et conditions aux limites artificielles en mecanique des fluides: methode optimisee d’Ordre 2, These de Doctorat, Universite Paris XIII, 1998; F. Nataf, F. Rogier, M3 AS 5 (1) (1995) 67–93]. Numerical tests illustrate the efficiency of the method.

A new cement to glue non-conforming grids with Robin interface conditions: The finite volume case

Robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence. Non conforming grids make the grid generation much easier and faster since it is then a parallel task. The goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of Robin interface conditions with non-matching grids. We consider both a symmetric definite positive operator and the convection-diffusion equation discretized by finite volume schemes. Numerical results are shown.

Algorithm 932: PANG: Software for nonmatching grid projections in 2D and 3D with linear complexity

We design and analyze an algorithm with linear complexity to perform projections between 2D and 3D nonmatching grids. This algorithm, named the PANG algorithm, is based on an advancing front technique and neighboring information. Its implementation is surprisingly short, and we give the entire Matlab code. For computing the intersections, we use a direct and numerically robust approach. We show numerical experiments both for 2D and 3D grids, which illustrate the optimal complexity and negligible overhead of the algorithm. An outline of this algorithm has already been presented in a short proceedings paper of the 18th International Conference on Domain Decomposition Methods (see Gander and Japhet [2008]).

An Algorithm for Non-Matching Grid Projections with Linear Complexity

Non-matching grids are becoming more and more common in scientific computing. Examples are the Chimera methods proposed by [20] and analyzed in [2], the mortar methods in domain decomposition by [1], and the patch method for local refinement by [6], and [17], which is also known under the name ’numerical zoom’, see [9]. In the patch method, one has a large scale solver for a particular partial differential equation, and wants to add more precision in certain areas, without having to change the large scale code. One thus introduces refined, possibly non-matching patches in these regions, and uses a residual correction iteration between solutions on the patches and solutions on the entire domain, in order to obtain a more refined solution in the patch regions. The mortar method is a domain decomposition method that permits an entirely parallel grid generation, and local adaptivity independently of neighboring subdomains, because grids do not need to match at interfaces. The Chimera method is also a domain decomposition method, specialized for problems with moving parts, which inevitably leads to non-matching grids, if one wants to avoid regridding at each step. Contact problems in general lead naturally to nonmatching grids. In all these cases, one needs to transfer approximate solutions from one grid to a non-matching second grid by projection. This operation is known in the literature under the name mesh intersection problem in [12], intergrid communication problem in [16], grid transfer problem in [18], and similar algorithms are also needed when one has to interpolate discrete approximations, see [13, Chap. 13].

Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation

In many fields of applications such as reactive transport or ocean-atmosphere coupling, models with very different spatial and time scales have to be coupled. Optimized Schwarz Waveform Relaxation methods (OSWR), applied to linear advection-reaction-diffusion problems in [1, 8], provide efficient solvers for this purpose. They have two main advantages: first, they are global in time and thus permit non conforming space-time discretization in different subdomains, and second, few iterations are needed to compute an accurate solution, due to optimized transmission conditions. It has been proposed in [4] to use a discontinuous Galerkin method in time as a subdomain solver. Rigorous analysis can be made for any degree of accuracy and local time-stepping, and finally time steps can be adaptively controlled by a posteriori error analysis, see [6, 7, 10].

Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations

This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem. Two approaches are considered: one uses the time-dependent Steklov--Poincare operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and different time grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness of the Robin subdomain problems involved in the OSWR method and a convergence proof of the OSWR algorithm are given for the mixed formulation. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.

Advection Diffusion Problems with Pure Advection Approximation in Subregions

Martin J. Gander, Laurence Halpern, Caroline Japhet, and Veronique Martin 1 Universite de Geneve, 2-4 rue du Lievre, CP 64, CH-1211 Geneve, Switzerland. martin.gander@math.unige.ch 2 LAGA, Universite Paris XIII, 99 Avenue J.-B. Clement, 93430 Villetaneuse, France. {halpern,japhet}@math.univ-paris13.fr 3 LAMFA UMR 6140, Universite Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens Cedex 1, France. veronique.martin@u-picardie.fr

Optimized Schwarz Waveform Relaxation and Discontinuous Galerkin Time Stepping for Heterogeneous Problems

We design and analyze a Schwarz waveform relaxation algorithm for domain decomposition of advection-diffusion-reaction problems with strong heterogeneities. The interfaces are curved, and we use optimized Ventcell transmission conditions. We analyze the semidiscretization in time with discontinuous Galerkin as well. We also show two-dimensional numerical results using generalized mortar finite elements in space.

Optimized Schwarz Waveform Relaxation Algorithms with Nonconforming Time Discretization for Coupling Convection-diffusion Problems with Discontinuous Coefficients

We present and study an optimized Schwarz Waveform Relaxation algorithm for convection-diffusion problems with discontinuous coefficients. Such analysis is a first step towards the coupling of heterogeneous climatic models. The SWR algorithms are global in time, and thus allow for the use of non conforming space-time discretizations. They are therefore well adapted to coupling models with very different spatial and time scales, as in ocean-atmosphere coupling. As the cost per iteration can be very high, we introduce new transmission conditions in the algorithm which optimize the convergence speed. In order to get higher order schemes in time, we use in each subdomain a discontinuous Galerkin method for the time-discretization. We present numerical results to illustrate this approach, and we analyse numerically the time-discretization error.

Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation for Heterogeneous Problems

We consider the question of domain decomposition for evolution problems with discontinuous coefficients. We design a method relying on four ingredients: extension of the optimized Schwarz waveform relaxation algorithms as described in [1], discontinuous Galerkin methods designed in [7], time windows, and a generalization of the projection procedure given in [6]. We so obtain a highly performant method, which retains the approximation properties of the discontinuous Galerkin method. We present numerical results, for a two-domains splitting, to analyze the time-discretization error and to illustrate the efficiency of the DGSWR algorithm with many time windows. This analysis is in continuation with the approach initiated in DD16 [2, 5], with applications in climate modeling, or nuclear waste disposal simulations.