Past Event: Oden Institute Seminar

Fast high-order solvers on the simplicial de Rham complex via sparsity-promoting bases

Dr. Pablo Brubeck Martinez, Mathematical Institute, University of Oxford

Abstract

We present new high-order finite elements discretizing the L2 de Rham complex on triangular and tetrahedral meshes. The finite elements discretize the same spaces as usual, but with different basis functions. They allow for fast linear solvers based on static condensation and space decomposition methods. The new elements build upon the definition of degrees of freedom for interpolation given by Demkowicz et al. [1], and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the L2- and H(d)-inner products (d = grad, curl, or div). On the reference symmetric simplex, the resulting stiffness matrix has diagonal interior block, and does not couple together the interior and interface degrees of freedom. Thus, on the reference simplex, the Schur complement resulting from elimination of interior degrees of freedom is simply the interface block itself. 

This sparsity is not preserved on arbitrary cells mapped from the reference cell. Nevertheless, the interior-interface coupling is weak because it is only induced by the geometric transformation. We devise a preconditioning strategy by neglecting the interior-interface coupling.  We precondition the interface Schur complement with the interface block, and simply apply point-Jacobi to precondition the interior block. We further precondition the interface block by applying a space decomposition method with small subdomains constructed around vertices, edges, and faces. This allows us to solve the canonical Riesz maps in H(grad), H(curl), and H(div), at very high order.  We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.

Biography

Pablo Brubeck is a postdoctoral research associate at the Mathematical Institute at the University of Oxford, UK, from where he also acquired his doctoral degree in 2023. His research focuses on fast solvers for high-order finite element discretizations using multigrid and domain-decomposition techniques, and his interests span across numerical analysis, scientific computing, and computational Physics. He also contributes to the development of the Firedrake finite element library.