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

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Qk 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 R2. 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.