非协调有限元方法的逐点局部后验误差估计
Localized pointwise a posteriori error estimates for nonconforming finite element methods
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中文总结 AI 辅助
针对泊松和双调和方程的非协调有限元离散，本文设计新型权重函数，推导Crouzeix--Raviart方法及Morley离散的局部后验误差估计，实现对相关误差的精确控制。
中文摘要 AI 辅助
本文针对泊松方程和双调和方程的非协调有限元离散，建立了逐点局部后验误差估计。针对泊松问题，推导了Crouzeix--Raviart方法的函数值与破碎梯度误差的局部估计；针对双调和方程的Morley离散，推导了控制局部Hessian误差的局部后验估计。关键是设计了两个新型权重函数，用于对双调和方程正则格林函数导数的\boldsymbol{L^1}范数进行精确估计。
英文摘要
This paper establishes the first localized pointwise a posteriori error estimates for nonconforming finite element discretizations of the Poisson and biharmonic equations. For the Poisson problem, we derive localized estimates for the function-value and broken gradient errors of the Crouzeix--Raviart method. For the Morley discretization of the biharmonic equation, we derive a localized a posteriori estimate that controls the local Hessian error. This provides the first pointwise a posteriori error analysis for the biharmonic equation, for either conforming or nonconforming finite element methods. The key ingredient, absent from pointwise analysis of second order PDEs, is the design of two novel weight functions that facilitate sharp estimates of the $L^1$ norms of derivatives of a regularized Green's function for the biharmonic operator.
