Mesh-Independent Operator Preconditioning for Boundary Elements on Open Curves
Mesh-Independent Operator Preconditioning for Boundary Elements on Open Curves
复制标题
开曲线上边界元的网格独立算子预处理
DOI：
10.1137/130947040
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
C. Urzúa
中科院分区：
文献类型：
--
作者：
R. Hiptmair;C. Jerez;C. Urzúa
Poisson方程在有界Lipschitz曲线$\mathcal{C}$外部的边值问题可以被改写为第一类边界积分方程，其特征是弱奇异或超奇异边界积分算子（BIO）。基于[C. Jerez-Hanckes和J. Nedeigh，SIAM J. Math. Anal.，44（2012），pp. 2666--2694]对于$\mathcal{C}=[-1，1]$，这些生物学对象的逆，我们通过零阶和一阶边界元对Galerkin-Petrov离散化产生的线性方程组进行算子预处理。预条件依赖于边界元空间上定义的双重网格，他们可以表现出一致的独立的自由度，即使是家庭的局部细化网格的数量。
Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve $\mathcal{C}$ can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIOs). Based on the recent discovery in [C. Jerez-Hanckes and J. Nedelec, SIAM J. Math. Anal., 44 (2012), pp. 2666--2694] of inverses of these BIOs for $\mathcal{C}=[-1,1]$, we pursue operator preconditioning of the linear systems of equations arising from Galerkin--Petrov discretization by means of zeroth- and first-order boundary elements. The preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes.