Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

TLDR

This paper considers multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions with substantial robustness with respect to the dimensionality d and to the polynomial degree k.

Abstract

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div − a ( x ) ∇ · , with a continuous and positive over Ω ¯ , Ω being an open and bounded subset of R 2 . While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d ≥ 2 , showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k.