Computing the Distance between Two Finite Element Solutions Defined on Different 3D Meshes on a GPU

This article introduces a new method to efficiently compute the distance (i.e., $L^p$ norm of the difference) between two functions supported by two different meshes of the same 3D domain. The functions that we consider are typically finite element solutions discretized in different function spaces supported by meshes that are potentially completely unrelated. Our method computes an approximation of the distance by resampling both fields over a set of parallel 2D regular grids. By leveraging the parallel horse power of computer graphics hardware (graphics processing unit (GPU)), our method can efficiently compute distances between meshes with multimillion elements in seconds. We demonstrate our method applied to different problems (distance between known functions, Poisson solutions, and linear elasticity solutions) using different function spaces (Lagrange polynomials from order one to seven) and different meshes (tetrahedral and hexahedral, with linear or quadratic geometry).