PCPATCH

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.

FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.

COARSE SPACES BY ALGEBRAIC MULTIGRID: MULTIGRID CONVERGENCE AND UPSCALING ERROR ESTIMATES

We give an overview of a number of algebraic multigrid methods targeting finite element discretization problems. The focus is on the properties of the constructed hierarchy of coarse spaces that guarantee (two-grid) convergence. In particular, a necessary condition known as "weak approximation property," and a sufficient one, referred to as "strong approximation property," are discussed. Their role in proving convergence of the TG method (as iterative method) and also on the approximation properties of the algebraic mottigrid (AMG) coarse spaces if used as discretization tool is pointed out. Some preliminary numerical results illustrating the latter aspect are also reported.