Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk

François Alouges, Martin Averseng
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引用次数: 2

Abstract

We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in $\R^3$.  Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolutions and can thus be evaluated cheaply in the context of iterative methods.       For the Laplace equation (i.e. for the wavenumber $k=0$) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of $k$ and on locally refined meshes.
圆盘上亥姆霍兹第一类积分方程的准局部和频率稳健预处理器
我们针对亥姆霍兹散射问题提出了$\R^3$中平面圆盘形屏幕的预处理。 这些先决条件器是作用于屏幕的某些偏微分算子的平方根近似值。它们的矩阵向量乘积只涉及几个稀疏的系统分辨率,因此可以在迭代法中便宜地进行评估。 对于在圆盘和规则网格上具有迪里夏特条件的拉普拉斯方程(即波长为 $k=0$),我们证明了预条件线性系统在网格大小上具有均匀的有界条件数。我们还进一步提供了数值证据,表明预条件器在 $k$ 的大值和局部细化网格上也表现良好。
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