论文标题
对紧凑扰动边界积分方程的最小二乘过采样搭配方法的分析
An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions
论文作者
论文摘要
在最近的工作中（Maierhofer＆Huybrechs，2022年，Adv。Comput。Math。），作者表明，最小二乘可以过采样可改善与某些伪分化形式的运算符的边界积分方程的搭配方法的收敛性能。基本原则是，离散方法在适当的意义上近似于bubnov $ - $ galerkin方法。在目前的工作中，我们将整个运算符被紧凑型操作员$ \ Mathcal {k} $扰动时，将此分析扩展到了，它是边界上的sobolev space上的地图连续的，$ \ mathcal {k}：h^{p} \ rightArrow h^{q} {q} {q} $ p，for $ p，p，q n $ 这项研究使得离散的Bubnov-Galerkin正交条件中的测试和试验在不受干扰的设置中都经过修改，这一事实使这项研究变得复杂。我们的分析保证，在更一般的情况下，保留了有关最佳收敛速率和足够过度采样率的先前结果。确实，这是第一次，该分析对在2D中任意平滑的Jordan曲线上的边界积分公式的边界积分公式的最小二乘搭配的优势进行了完整的解释。我们的理论结果与数值实验非常吻合。
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.
