Solvers for mixed finite element problems using Poincar\'e operators based on spanning trees

TL;DR

This paper introduces a new explicit construction of Poincaré operators using spanning trees for finite element spaces, leading to efficient solvers for Hodge-Laplace problems with robust preconditioning.

Contribution

It presents a novel explicit method for constructing Poincaré operators based on spanning trees, enabling stable decompositions and efficient preconditioned solvers for mixed finite element problems.

Findings

The proposed solver solves four smaller symmetric positive definite systems.

Numerical experiments confirm the efficiency and robustness of the method.

The approach improves computational performance for elliptic mixed finite element problems.

Abstract

We propose an explicit construction of Poincar\'e operators for the lowest order finite element spaces, by employing spanning trees in the grid. In turn, a stable decomposition of the discrete spaces is derived that leads to an efficient numerical solver for the Hodge-Laplace problem. The solver consists of solving four smaller, symmetric positive definite systems. We moreover place the decomposition in the framework of auxiliary space preconditioning and propose robust preconditioners for elliptic mixed finite element problems. The efficiency of the approach is validated by numerical experiments.