We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.

中文翻译：

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奇异源泊松问题数值逼近的加权设置

我们考虑 Poisson 类型问题的近似，其中源由奇异度量给出并且域是凸多边形或多面体域。首先，我们证明了当源属于加权 Sobolev 空间的对偶时 Poisson 问题的适定性，其中权重属于 Muckenhoupt 类。其次，我们证明了在网格族准均匀假设下标准有限元近似的加权范数的稳定性。