A spectral element method for meshes with skinny elements

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.