PCPATCH: software for the topological construction of multigrid relaxation methods

Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gauß–Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this article, 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.