PCPATCH: software for the topological construction of multigrid relaxation methods

TL;DR

PCPATCH is a flexible software framework integrated with PETSc that enables rapid development of multigrid relaxation methods through topological space decompositions, improving solver efficiency for complex equations.

Contribution

The paper introduces PCPATCH, a unifying software abstraction for topological construction of multigrid relaxation schemes, adaptable via solver options for various complex problems.

Findings

Enables rapid development of multigrid solvers

Supports complex equations with semidefinite or saddle point structures

Implemented in PETSc for ease of use

Abstract

Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gau{\ss}-Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this paper we present a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods. Space decompositions are specified by collecting topological entities in a mesh (such as all vertices or faces) and applying a construction rule (such as taking all degrees of freedom in the cells around each entity). The software is implemented in PETSc and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. In turn, this allows for the very rapid development of fast solvers for difficult problems.