Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.

中文翻译：

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泊松方程在三维域边缘的奇异解及其预测－校正有限元方法的处理

带有边的三维域中的边值问题的解可能会表现出奇异性，这会影响有限元解的精度以及误差估计的收敛速度。本文考虑对典型域泊松方程边界值问题Ω⊂ℝ ³具有边缘奇异性，一方面提出了通量强度函数的显式计算公式。另一方面，它提出并分析了规则网格上的非协调有限元方法，以有效地处理奇点。本方法的新颖之处在于，在定义有限元逼近的后处理过程中，使用了用于磁通强度函数的显式公式。H ¹（Ω）中的先验误差估计表明，本算法表现出与常规解问题相同的收敛速度。