A Mixed Finite Element Method for the Dirichlet Vector Laplacian in Three Dimensions

TL;DR

This paper develops a mixed finite element method for the 3D vector Laplacian with Dirichlet boundary conditions, providing error estimates and extending previous 2D analysis to 3D domains with general topology.

Contribution

It introduces a novel finite element approach for the 3D vector Laplacian with Dirichlet conditions, including error analysis and extension of 2D results to 3D.

Findings

Achieved $(k-1/2)$-th order convergence in energy norm on general domains.

Achieved $k$-th order convergence in $L^2$ norm on convex domains.

Extended 2D analysis to 3D domains with general topology.

Abstract

This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The Dirichlet condition disrupts the structure of the standard de Rham complex, requiring the vorticity to be sought in a non-standard function space to achieve well-posedness. We derive error estimates that confirm the numerically observed suboptimal convergence rates. In particular, by developing a discrete Caccioppoli-type inequality for discrete curl-harmonic functions, we prove $(k-1/2)$-th order convergence in the energy norm on general domains and $k$-th order convergence in $L^2$ on convex domains, where $k \ge 1$ denotes polynomial degrees of the finite element spaces. These results extend the previous two-dimensional analysis developed in [Arnold, D.N., Falk, R.S. and Gopalakrishnan, J., 2012. Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions. Mathematical Models and Methods in Applied Sciences, 22(9), p.1250024.]~to three-dimensional domains with general topology. As a direct application, a discretization of the Stokes problem in vorticity-velocity-pressure form is studied.