Bridging the gap between geometric and algebraic multi-grid methods

Abstract

In this paper, a multi-grid solver for the discretisation of partial differential equations on complicated domains is developed. The algorithm requires as input the given discretisation only instead of a hierarchy of discretisations on coarser grids. Such auxiliary grids and discretisations are generated in a black-box fashion and are employed to define purely algebraic intergrid transfer operators. The geometric interpretation of the algorithm allows one to use the framework of geometric multigrid methods to prove its convergence. The focus of this paper is on the formulation of the algorithm and the demonstration of its efficiency by numerical experiments, while the analysis is carried out for some model problems.