SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1117-1139, January 2021.
In this work we consider the dual-mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is nonstandard, as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher- and lower-regularity terms. The lower-regularity terms are here explicitly known and capture the solution singularities. The higher-regularity terms, meanwhile, are defined as the solution of an associated mixed Poisson equation. With the solution splitting in hand, we then define a singularity removal--based mixed finite element method in which only the higher-regularity terms are approximated numerically. This method yields a significant improvement in the convergence rate when compared to approximating the full solution. In particular, we show that the singularity removal--based method yields optimal convergence rates for lowest-order Raviart--Thomas and discontinuous Lagrange elements.

中文翻译：

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线源泊松问题的混合方法

SIAM数值分析学报，第59卷，第2期，第1117-1139页，2021年1月。
在这项工作中，我们考虑了带有线源的泊松方程的双重混合变分公式。这个问题的分析和近似是非标准的，因为线源导致解是奇异的。首先，我们证明该问题允许在适当加权的Sobolev空间中求解。接下来，我们表明，在对问题参数进行一些假设的情况下，该解决方案允许将分解分为较高和较低规则性的术语。此处，较低规则性的术语是明确已知的，并且捕获了解决方案的奇点。同时，高正则性项定义为关联的混合泊松方程的解。通过拆分解决方案，我们然后定义了基于奇点去除的混合有限元方法，其中仅对较高正则性项进行数值近似。与逼近完整解决方案相比，该方法可显着提高收敛速度。特别是，我们证明了基于奇异点去除的方法对于最低阶Raviart-Thomas和不连续的Lagrange元素产生了最优的收敛速度。