A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems
A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems
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高度不定变分问题的精化伽辽金误差和稳定性分析
DOI：
10.1137/060654177
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
S. Sauter
中科院分区：
文献类型：
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作者：
L. Banjai;S. Sauter
最近，第二作者介绍了一种用于高度不确定Helmholtz问题的精细有限元分析。我们推广的分析应用到一个抽象的高度不定变分问题的Galerkin方法。在精细分析中，稳定性和准最优误差估计的条件表示为近似性质${\cal T}（S）\approx S$和${\cal T}（u+S）\approx S$。这里，$u$是原变分问题的解，${\cal T}$是某个连续解算子，$S$是有限维的测试和试探空间。这种抽象分析方法可用于高频亥姆霍兹问题的有限元和边界元解。我们应用分析调查的Brakhage-Werner边界积分公式的Helmholtz问题，离散化的标准Galerkin边界元法的属性。在散射的情况下，由单位球，我们推导出明确的依赖性的错误和波数$k$的稳定性条件。我们证明了$hk \lesssim 1$是稳定的充分条件和拟最优误差估计。此外，我们表明，常数quasioptimality是独立的$k$，这是一个改进以前可用的结果。因此，边界元法不受污染影响。
Recently, a refined finite element analysis for highly indefinite Helmholtz problems was introduced by the second author. We generalize the analysis to the Galerkin method applied to an abstract highly indefinite variational problem. In the refined analysis, the condition for stability and a quasi-optimal error estimate are expressed in terms of approximation properties ${\cal T}(S) \approx S$ and ${\cal T}(u+S) \approx S$. Here, $u$ is the solution of the original variational problem, ${\cal T}$ is a certain continuous solution operator, and $S$ is the finite dimensional test and trial space. The abstract analysis can be applied to both finite and boundary element solutions of high-frequency Helmholtz problems. We apply the analysis to investigate the properties of the Brakhage-Werner boundary integral formulation of the Helmholtz problem, discretized by a standard Galerkin boundary element method. In the case of scattering by the unit sphere, we derive the explicit dependence of the error and of the stability condition on the wave number $k$. We show that $hk \lesssim 1$ is a sufficient condition for stability and a quasi-optimal error estimate. Further, we show that the constant of quasioptimality is independent of $k$, which is an improvement over previously available results. Thus, the boundary element method does not suffer from the pollution effect.