Fast solvers for high-order mixed FEMs

Marcus Sarkis (Department of Mathematical Sciences, Worcester Polytechnic Institute, USA)

Abstract. In this work, we propose iterative schemes to solve saddle point problems arising in the context of mixed finite elements. We focus the talk on the $RT_k$ Raviart Thomas of order $k$ discretization for the transmission problem. The strategy starts by applying the technique developed by Devloo, Fernandes, Gomes, Orlandini and Shauer 2022-CMAME where $RT_k$ divergence free basis functions for the whole region can be constructed locally in each element based on edges and interior degrees of fredom is each element. This freediv subpace is augmented with $RT_0$ in order to contains all the divfree subspace of the $RT_k$. We then develop three preconditioners which map to divfree subpaces and so the conjugated gradient method can be applied. We present theoretical results and numerical experiments.
This is a work in collaboration with Philippe Devloo and Sonia Gomes from Unicamp-Brazil and Jeferson Fernandes from UFMG-Brazil.