A priori error estimates of regularized elliptic problems

TL;DR

This paper establishes convergence rates for regularized approximations of the Dirac delta in Sobolev norms and applies these results to improve error estimates in finite element methods for elliptic problems with singular sources.

Contribution

It provides the first a priori convergence rates for Dirac delta approximations in Sobolev norms with minimal regularity assumptions, and applies these to enhance error estimates in regularized immersed interface methods.

Findings

Convergence rates of Dirac delta approximations in Sobolev norms are established.

Finite element methods achieve optimal error rates when using the regularized approach.

Numerical experiments confirm the theoretical error estimates.

Abstract

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method result in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.