Fast High-Order Finite Element Solvers on Simplices

We present new high-order finite elements discretizing the $L^2$ 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 given by (Demkowicz et al., De Rham diagram for $hp$ finite element spaces. Comput.~Math.~Appl., 39(7-8):29--38, 2000.), and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\}$).

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.

The combination of this approach with a space decomposition method on small subdomains constructed around vertices, edges, and faces allows us to efficiently solve the canonical Riesz maps in $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$, at very high order. We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.