MINRES for Second-Order PDEs with Singular Data

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