Localized pointwise error estimates for mixed finite element methods
Localized pointwise error estimates for mixed finite element methods
复制标题
混合有限元方法的局部逐点误差估计
作者：
A. Demlow
在本文中，我们给出了加权或局部逐点误差估计，该估计对于一般二阶线性椭圆问题的两种不同混合有限元方法以及单纯网格混合元素的一般选择是有效的。这些估计与 Schatz 最近为椭圆问题的基本 Galerkin 有限元方法所证明的估计类似，表明标量和向量变量中的点误差对解的导数的依赖性在性质上大多是局部的，或者相反，点误差的全局依赖性很弱。对于高阶元素，这种定位更加明显。我们的估计表明，除非使用最低阶 Brezzi-DouglasMarini 元素，否则会发生局部化，并且我们提供的计算示例表明，当使用这些元素时，误差确实没有局部化。
In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character, or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-DouglasMarini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.