Véronique Martin

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

An Asymptotic Approach to Compare Coupling Mechanisms for Different Partial Differential Equations

In many applications the viscous terms become only important in parts of the computational domain. A typical example is the flow of air around the wing of an airplane. It can then be desirable to use an expensive viscous model only where the viscosity is essential for the solution and an inviscid one elsewhere. This leads to the interesting problem of coupling partial differential equations of different types.

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 in one spatial dimension, 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. We illustrate our results with numerical experiments in one and two spatial dimensions.

How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions

The coupling of different types of partial differential equations is an active field of research, since the need for such coupling arises in various applications. A first main area is the simulation of complex objects, composed of different materials, which are naturally modeled by different equations; fluid-structure interaction is a typical example. A second main area is when homogeneous objects are simulated, but the partial differential equation modeling the object is too expensive to solve over the entire object. A simpler, less expensive model would suffice in most of the object to reach the desired accuracy. Fluid flow around an airplane could serve as an example, where viscous effects are important close to the airplane, but can be neglected further away. A third emerging area is the coupling of equations across dimensions, for example the blood flow in the artery can be modeled by a one dimensional model, but in the heart, it needs to be three dimensional.

Multiscale analysis of heterogeneous domain decomposition methods for time-dependent advection–reaction–diffusion problems

Abstract Domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods. We are interested here in the case where there is an accurate but expensive model one should use in the entire domain, but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected. For the model problem of a time dependent advection–reaction–diffusion equation in one spatial dimension, we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain. Using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain, we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non-variational heterogeneous domain decomposition methods. We show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis, and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity.