MINRES for second-order PDEs with singular data

TL;DR

This paper develops regularization techniques for minimum residual methods like DPG and least-squares FEM to handle singular data such as non square-integrable loads in second-order PDEs, with proven convergence and numerical validation.

Contribution

It introduces regularization strategies enabling DPG and least-squares FEM to effectively handle singular data in second-order PDEs, extending their applicability.

Findings

Proven convergence orders for regularized methods with singular data

Numerical experiments confirm theoretical convergence results

Approach applicable to general well-posed second-order problems

Abstract

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.