二维紧扰动边界积分方程的最小二乘过采样配置法分析
An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions
摘要：
在最近的工作中（Maierhofer & Huybrechs，2022，Adv.数学），作者证明了最小二乘过采样可以改善包含某些伪微分形式算子的边界积分方程的配置法的收敛性质。其基本原理是离散方法在适当的意义上近似Bubnov $-$ Galerkin方法。在目前的工作中，我们将这种分析的情况下，当积分算子扰动的紧算子$\mathcal{K}$是连续的Sobolev空间上的边界上的映射，$\mathcal{K}:H^{p}\rightarrow H^{q}$的所有$p,q\in\mathbb{R}$。这项研究是复杂的事实，测试和测试函数在离散Bubnov-Galerkin正交条件下修改的未扰动设置。我们的分析保证，以前的结果有关的最佳收敛速度和足够的过采样率被保存在更一般的情况下。事实上，第一次，这种分析提供了一个完整的解释的优势，最小二乘过采样配置的边界积分公式的拉普拉斯方程在任意光滑的约旦曲线在2D。我们的理论结果与数值实验结果吻合得很好。
英文摘要：
In recent work (Maierhofer & Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods for boundary integral equations involving operators of certain pseudo-differential form. The underlying principle is that the discrete method approximates a Bubnov$-$Galerkin method in a suitable sense. In the present work, we extend this analysis to the case when the integral operator is perturbed by a compact operator $\mathcal{K}$ which is continuous as a map on Sobolev spaces on the boundary, $\mathcal{K}:H^{p}\rightarrow H^{q}$ for all $p,q\in\mathbb{R}$. This study is complicated by the fact that both the test and trial functions in the discrete Bubnov-Galerkin orthogonality conditions are modified over the unperturbed setting. Our analysis guarantees that previous results concerning optimal convergence rates and sufficient rates of oversampling are preserved in the more general case. Indeed, for the first time, this analysis provides a complete explanation of the advantages of least-squares oversampled collocation for boundary integral formulations of the Laplace equation on arbitrary smooth Jordan curves in 2D. Our theoretical results are shown to be in very good agreement with numerical experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
