Wietse Boon: Solvers for mixed finite element problems based on spanning trees

Abstract:

Mixed finite element methods are capable of imposing physical conservation laws at the discrete level by preserving the structure of governing PDEs. However, this often requires a large number of degrees of freedom and leads to computationally demanding saddle point systems. In this talk, we consider a decomposition of the lowest order finite element spaces based on spanning trees in the grid. This decomposition allows us to derive so-called Poincaré operators for the finite element de Rham complex. Using this abstract tool, we form a new basis in which the (Hodge-)Laplace problem unravels from a large saddle-point problem into four smaller, symmetric positive definite systems. In turn, we achieve a significant computational speed-up, without loss of accuracy, for applications ranging from flow in porous media, to electromagnetics and solid mechanics. We highlight two additional, practical implications of the theory. First, it allows us to construct robust preconditioners for general, elliptic finite element problems. Additionally, we use the tools to develop neural network solvers with guaranteed conservation of mass or momentum.