A spectral element method for meshes with skinny elements

TL;DR

This paper introduces a spectral element method that remains numerically stable on meshes with skinny elements, enabling high-degree polynomial discretizations for PDEs, demonstrated through Navier--Stokes simulations.

Contribution

The authors develop a novel spectral element method that is stable on meshes with skinny elements, expanding the applicability of spectral methods to anisotropic meshes.

Findings

Method is stable on meshes with skinny elements.

Allows high-degree polynomial discretizations.

Successfully applied to Navier--Stokes simulations.

Abstract

When numerically solving partial differential equations (PDEs), the first step is often to discretize the geometry using a mesh and to solve a corresponding discretization of the PDE. Standard finite and spectral element methods require that the underlying mesh has no skinny elements for numerical stability. Here, we develop a novel spectral element method that is numerically stable on meshes that contain skinny elements, while also allowing for high degree polynomials on each element. Our method is particularly useful for PDEs for which anisotropic mesh elements are beneficial and we demonstrate it with a Navier--Stokes simulation. Code for our method can be found at <https://github.com/ay2718/spectral-pde-solver>.