各向异性网格下非协调元的后验误差估计

A Posteriori Error Estimators of Anisotropic Nonconforming Finite Element

在各向异性网格下,给出了泊松方程的非协调有限元逼近的残量型后验误差估计.由于直接采用各向异性网格剖分比各向同性网格能在很大程度上节省自由度和提高计算精度,但会使后验误差估计中出现一个无界的因子,本文通过引入匹配函数来反映各向异性网格与函数的匹配程度,从而避免了这个因子的出现.非协调元因其很好的收敛性而有相当好的应用价值.利用Helmholtz分解和误差的正交性对非协调元引起的相容项进行处理,得到了误差的上界,证明了估计子的可靠性.

A posteriori residual error estimators for nonconforming finite element approximation to Poisson problem were given on anisotropic meshes.Although the anisotropic meshes could save the freedom and improve the accuracy of calculation greatly compared to the isotropic ones,but there would appear an unbounded factor in the a posteriori error estimation.In order to avoid the factor,a matching function was defined to measure whether the anisotropic mesh was well aligned with the anisotropic solution.As the nonconforming finite element has very good convergence,it was a wide application.For the nonconforming item,the technique was based on a Helmholtz decomposition with some orthogonal relations for the error,and the upper error bound was proved,thus proving the reliability of the estimator.