论文标题
在混合的fem和fosls上，带有$ h^{ - 1} $ loads
On a mixed FEM and a FOSLS with $H^{-1}$ loads
论文作者
论文摘要
我们研究了混合有限元方法（混合FEM）和一阶系统最小二乘有限元（FOSLS）的变体，用于泊松问题，我们通过合适的正则化替换负载，允许使用$ h^{ - 1} $ loads。我们证明，任何有限的$ h^{ - 1} $投影仪都可以用于定义正则化，并产生最低级混合fem resp的准选项。较弱的规范中的fosl。给出了构建此类投影仪的示例。一个基于加权clément准互式polator的伴随。我们证明该clément操作员具有二阶近似属性。对于修改的混合方法，我们在最小的规律性假设下显示了后处理溶液的最佳收敛速率 - 这是对没有正则化的最低级混合FEM无效的结果。数值示例总结了这项工作。
We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use $H^{-1}$ loads. We prove that any bounded $H^{-1}$ projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Clément quasi-interpolator. We prove that this Clément operator has second-order approximation properties. For the modified mixed method we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions -- a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.
