In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of π/2 or 3π/2, we have obtained an explicit a prioriH01 error estimation ofO(h) and anL2 error estimation ofO(h2) for such a finite element solution of the Poisson equation. Particularly, we emphasize that all constants in our error estimates are numerically determined, which plays an essential role in the numerical verification of solutions to non-linear elliptic problems.

中文翻译：

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使用奇异函数的非凸域有限元离散化的构造性先验误差估计

在具有可重入角的有界域中用有限元方法求解椭圆问题时，可以通过在通常的插值基上添加奇异函数来提高收敛速度。当域被形成 π/2 或 3π/2 角的线段包围时，我们已经获得了 O(h) 的显式先验 H01 误差估计和 O(h2) 的 L2 误差估计泊松方程。特别是，我们强调误差估计中的所有常数都是数值确定的，这在非线性椭圆问题解的数值验证中起着至关重要的作用。