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.

翻译：最低残留方法,如最小方位限制元素法(FEM)或具有最佳测试功能的不连续Petrov-Galerkin方法(DPG)通常不包括单项数据,例如非可方可容物负荷。我们认为Poisson问题采用DPG方法和最小方块FEM。两种方法都分析允许使用$H ⁇ -1美元负荷的正规化方法,并研究点负荷案例。对于所有案例,我们证明是适当的趋同命令。我们提出了各种数字实验,以证实我们的理论结果。我们的方法延伸至一般的妥善第二顺序问题。