Résolution des PDE en parallèle

Question

J'ai lu différentes approches sur la façon de résoudre des pdes en parallèle qui sont discrétisées en utilisant la méthode des éléments finis. Par example:

1. Approche de décomposition de domaine sans chevauchement comme mentionné dans https://imsc.uni-graz.at/haasegu/Papers/Douglas-Haase-Langer/textbook.pdf au chapitre 5.2 . Chaque processus fonctionne sur chaque propre domaine et le vecteur de solution a un stockage cohérent tandis que les côtés droit, les résidus et la matrice de rigidité ont un stockage additif.

2. Répartissez le maillage en N parties. Chaque processeur a des informations sur son propre sous-domaine plus une couche fantôme (approche des solveurs globaux).

Quelles sont les différences entre ces deux méthodes, avantages et inconvénients? Quelle méthode de parallélisation est utilisée par les logiciels FEM?

Answer 1

Domain decomposition was developed in the late 1990s and early 2000s because it allowed the re-use of sequential PDE solvers: You only have to write a wrapper around it that sends the computed solution to other processors, receives other processors' solutions, and uses these as boundary values for the next iteration. This works reasonably well for the small numbers of processors that were used at the time (a few dozen to at most a few hundred), but the approach does not perform well with large numbers of processors.

The approach almost universally used today is the second method you outline, where we think of the mesh and the linear system as one global one; it just happens to be stored in a way that distributes the data to many processors. In other words, we don't decompose the problem into smaller problems, we just decompose the storage of the data associated with the one global problem. This has required a lot of software development in libraries such as PETSc, Trilinos, libMesh, or the deal.II project which I co-lead. But, on the upside, this perspective leads to methods that can be efficiently solved, and as a consequence, they have largely supplanted domain decomposition methods in the last fifteen or so years.