Analysis of A Mixed Finite Element Method for Poisson's Equation with Rough Boundary Data

TL;DR

This paper introduces a mixed finite element method that directly solves Poisson's equation with rough boundary data, achieving convergence rates that depend on domain convexity, and confirms these rates through numerical experiments.

Contribution

It develops a Raviart--Thomas mixed finite element approach for Poisson's equation with rough boundary data, avoiding regularization of boundary conditions.

Findings

Convergence rate of O(h^{1/2}) in convex domains.

Convergence rate of O(h^{s-1/2}) in nonconvex domains.

Numerical experiments confirm theoretical convergence rates.

Abstract

This paper is concerned with finite element methods for Poisson's equation with rough boundary data. Conventional methods require that the boundary data $g$ of the problem belongs to $H^{1/2} (\partial \Omega)$. However, in many applications one has to consider the case when $g$ is in $L^2(\partial \Omega)$ only. To this end, very weak solutions are considered to establish the well-posedness of the problem. Most previously proposed numerical methods use regularizations of the boundary data. The main purpose of this paper is to use the Raviart--Thomas mixed finite element method to solve the Poisson equation with rough boundary data directly. We prove that the solution to the proposed mixed method converges to the very weak solution. In particular, we prove that the convergence rate of the numerical solution is $O(h^{1/2})$ in convex domains and $O(h^{s-1/2})$ in nonconvex domains, where $s > 1/2$ depends on the geometry of the domain. The analysis is based on a regularized approach and a rigorous estimate for the corresponding dual problem. Numerical experiments confirm the theoretically predicted convergence rates for the proposed mixed method for Poisson's equation with rough boundary data.