A scalable and robust vertex-star relaxation for high-order FEM

TL;DR

This work tackles the difficulty of obtaining $p$-robust preconditioning for high-order finite element methods by introducing a scalable vertex-star relaxation that leverages a tensor-product basis to diagonalize interior blocks of patch matrices. A fast diagonalization (FDM) based, sparse interval basis renders patch matrices sparse and enables direct Cholesky factorization with complexity $\mathcal{O}(p^{3(d-1)})$ for factorization but $\mathcal{O}(p^{2(d-1)})$ per solve, making high-degree ($p$) computations feasible on Cartesian patches. The method is extended to non-Cartesian cells via a spectrally equivalent separable surrogate, ensuring robustness with respect to mesh size $h$ and polynomial degree $p$ for Poisson and, in mixed elasticity, $H(\mathrm{div})\times L^2$ formulations; results include 3D demonstrations up to $p=15$. Limitations include restricted applicability to problems with mixed derivatives and dependence on mesh quality; future work aims to replace the costly Cholesky with iterative patch solvers and algebraic multigrid techniques. Overall, the approach offers a practical path to scalable high-order preconditioning in structured, tensor-product-like settings, with potential extensions to broader coefficient variability.

Abstract

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We demonstrate the approach by solving the Poisson equation and a $H(\mathrm{div})$-conforming interior penalty discretization of linear elasticity in three dimensions at $p = 15$.

Figures (13)

• Figure 1: Subdomains for the additive Schwarz method on a regular mesh ($p=4$). In combination with a low-order coarse grid, the vertex-star patch gives a $p$-robust method for symmetric and coercive problems, while the cell-centered patch does not.
• Figure 1: Plots of the interval shape functions ($p=4$) and non-zero structure of the mass and stiffness matrices on the reference interval ($p=7$).
• Figure 1: Non-zero structure for the interior facet matrix $E^e$ on the interval ($p=4$).
• Figure 2: The FDM may be applied as a relaxation only on vertex-star patches that are structured, i.e. where the cells are laid out in a tensor product grid.
• Figure 2: Plots of the interval FDM shape functions ($p=4$) and non-zero structure of the mass and stiffness matrices in the FDM basis on the reference interval ($p=7$).

Theorems & Definitions (2)

• Theorem 1
• Proof 1