On the least-squares method
On the least-squares method
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DOI：
10.1016/s0045-7825(97)00192-8
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发表时间：
1998-01-01
影响因子：
7.2
通讯作者：
Jiang, BN
中科院分区：
文献类型：
--
作者：
Jiang, BN
证明了一般最小二乘法的理论基础是有界逆定理。这就解释了为什么在一个数学/计算框架内的最小二乘有限元法（LSFEM）无需任何特殊处理就可以为所有类型的偏微分方程提供数值解。建立了一般偏微分方程组和椭圆一阶系统的LSFEM误差估计。作为一个例子，考虑不可压缩的斯托克斯方程。一阶速度-压力-涡度公式中斯托克斯方程的主要部分由两个 div-curl 系统组成。该知识用于导出斯托克斯方程的非标准边界条件的所有允许组合，并表明在这些条件下，斯托克斯算子在完整的 H-1 意义上有界于下方，因此相应的 LSFEM 是最优的。还给出了数值结果来支持这一结论。
It is shown that the theoretical basis of the general least-squares method is the bounded inverse theorem. This explains why the least-squares finite element method (LSFEM) within one mathematical/computational framework without any special treatment can provide numerical solutions for all types of partial differential equations. Error estimates of LSFEM for general and elliptic first-order systems of partial differential equations are established. As an example, the incompressible Stokes equations are considered. The principal part of the Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems. This knowledge is employed to derive all permissible combinations of non-standard boundary conditions for the Stokes equations and to show that under these conditions the Stokes operator is bounded below in the full H-1 sense, and hence the corresponding LSFEM is optimal. The numerical results are also given to support this conclusion.