Abstract Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

中文翻译：

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多面体域椭圆方程各向异性有限元方法的先验分析

摘要 考虑具有混合边界条件的多面体域中的泊松方程。我们在通常的 Sobolev 空间 H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . 我们提出了以最优收敛速度逼近奇异解的各向异性有限元算法。特别是，我们的数值方法涉及几何约束较少但缺乏最大角度条件的各向异性渐变网格。这种网格上的最佳收敛通常需要纯狄利克雷边界条件。因此，我们结果的一个副产品是将这些各向异性网格的应用扩展到更广泛的实际计算，代价是获得“更平滑”的内部数据。