Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

TL;DR

This paper analyzes the error and convergence rates of a diffuse interface finite element method for solving elliptic Poisson problems with Dirichlet boundary conditions on embedded curved interfaces, providing theoretical and numerical validation.

Contribution

It provides a rigorous error analysis and convergence rates for a diffuse interface approach applied to elliptic problems with complex boundaries, including new duality techniques for $L^2$ estimates.

Findings

Error estimates in $H^1$, $L^2$, and $L^ Infinity$ norms in terms of layer width.

Convergence rates for finite element solutions considering layer width and mesh size.

Numerical experiments confirming the sharpness of the theoretical estimates.

Abstract

We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular domains. We estimate the errors introduced by these domain perturbations, and prove convergence and convergence rates in the $H^1$-norm, the $L^2$-norm and the $L^\infty$-norm in terms of the width of the diffuse layer. For an efficient numerical solution we consider the finite element method for which another domain perturbation is introduced. These perturbed domains are polygonal and non-convex in general. We prove convergence and convergences rates in the $H^1$-norm and the $L^2$-norm in terms of the layer width and the mesh size. In particular, for the $L^2$-norm estimates we present a problem adapted duality technique, which crucially makes use of the error estimates derived for the regularly perturbed domains. Our results are illustrated by numerical experiments, which also show that the derived estimates are sharp.