Nonlinear Nonoverlapping Schwarz Waveform Relaxation for Semilinear Wave Propagation

We introduce a non-overlapping variant of the Schwarz waveform relaxation algorithm for semilinear wave propagation in one dimension. Using the theory of absorbing boundary conditions, we derive a new nonlinear algorithm. We show that the algorithm is well-posed and we prove its convergence by energy estimates and a Galerkin method. We then introduce an explicit scheme. We prove the convergence of the discrete algorithm with suitable assumptions on the nonlinearity. We finally illustrate our analysis with numerical experiments.Comment: 20 page

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 Robin or Ventcell transmission conditions. We analyze the semi-discretization in time with Discontinuous Galerkin as well. We also show two-dimensional numerical results using generalized mortar finite elements in space

A new Algorithm Based on Factorization for Heterogeneous Domain Decomposition

Often computational models are too expensive to be solved in the entire domain of simulation, and a cheaper model would suffice away from the main zone of interest. We present for the concrete example of an evolution problem of advection reaction diffusion type a heterogeneous domain decomposition algorithm which allows us to recover a solution that is very close to the solution of the fully viscous problem, but solves only an inviscid problem in parts of the domain. Our new algorithm is based on the factorization of the underlying differential operator, and we therefore call it factorization algorithm. We give a detailed error analysis, and show that we can obtain approximations in the viscous region which are much closer to the viscous solution in the entire domain of simulation than approximations obtained by other heterogeneous domain decomposition algorithms from the literature.Comment: 23 page

Variational formulation for a nonlinear elliptic equation in a three-dimensional exterior domain

An existence result was obtained for a nonlinear second-order equation in an exterior domain of IR(3). The proof relies on a variational formulation in weighted Sobolev spaces

Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions

Optimized Schwarz Waveform Relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to a mathematically hard best approximation problem of homographic type. We study in this paper in detail this problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel type. We give for each case closed form asymptotic values for the parameters, which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, and we measure performance corresponding to our analysisComment: 42 page

HyperbolicBoundary Value Problemswith Trihedral Corners

International audienceExistence and uniqueness theorems are proved forboundary value problems with trihedral corners and distinct boundaryconditions on the faces. Part I treatsstrictly dissipative boundary conditions for symmetric hyperbolicsystems with elliptic or hidden elliptic generators. Part II treats the Bérengersplit Maxwell equations in three dimensions with possibly discontinuous absorptions. The discontinuity set of the absorptions or their derivatives has trihedral corners. Surprisingly, there is almost noloss of derivatives for the B\'erenger split problem. Bothproblems have their origins in numerical methods with artificialboundarie

Perfectly Matched Layers equations for 3D acoustic wave propagation in heterogeneous media

International audienceThis work is dedicated to the analysis of Berenger PML method applied to the 3D linearized Euler equations without advection terms, with variable wave velocity and acoustic impedance. It is an extension of a previous work presented in a 2D context [8]. The 3D linearized Euler equations are used to simulate propagation of acoustic waves beneath the subsurface. We propose an analysis of these equations in a general heterogeneous context, based on a priori error estimates. Following the method introduced by M ́ tral and Vacus [9], we derive an augmented system from the original one, involving the primitive unknowns and their first order spatial derivatives. We define a symetrizer for this augmented system. This allows to compute energy estimates in the three following cases: the Cauchy problem, the half-space problem with a non homogeneous Dirichlet boundary condition and finally the transmission problem between two half-spaces separated by an impedance discontinuity

A Schwarz Waveform Relaxation Method for Advection—Diffusion—Reaction Problems with Discontinuous Coefficients and Non-matching Grids

International audienceWe present a non-overlapping Schwarz waveform relaxation method for solving advection-reaction-diffusion problems in heterogeneous media. The do-main decomposition method is global in time, which permits the use of different time steps in different subdomains. We determine optimal non-local, and optimized Robin transmission conditions. We also present a space-time finite volume scheme es-pecially designed to handle such transmission conditions. We show the performance of the method on an example inspired from nuclear waste disposal simulations

The Analysis of Matched Layers

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of B\'erenger. We prove that the B\'erenger and closely related layers define well posed transmission problems in great generality. When the B\'erenger method or one of its close relatives is well posed, perfect matching is proved. The proofs use the energy method, Fourier-Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the B\'erenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in B\'erenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of B\'erenger's algorithm for Maxwell's equations our favorite choice leads to a strongly well posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence

Matching asymptotic method in propagation of cracks with Dugdale model

International audienceThe goal of this work is to apply the matching asymptotic method combined with a vari- ational approach to study the initiation and the propagation of a cohesive crack from the tip of a preexisting notch following the Dugdale cohesive force model when the characteristic length of the material (included in the Dugdale model) is small by comparison with the characteristic length of the body