Multiple traces boundary integral formulation for Helmholtz transmission problems
Multiple traces boundary integral formulation for Helmholtz transmission problems
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DOI：
10.1007/s10444-011-9194-3
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发表时间：
2012-07
影响因子：
1.7
通讯作者：
R. Hiptmair;C. Jerez-Hanckes
中科院分区：
文献类型：
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作者：
R. Hiptmair;C. Jerez-Hanckes
我们提出了一种新的边界积分公式的亥姆霍兹传输问题的有界复合散射体（即分段常数材料参数的“子域”），直接借给自己通过卡尔德龙投影算子预处理。该方法依赖于子域上的局部迹线和传输条件的弱执行。变分公式被设置在标准Dirichlet和特殊Neumann迹空间的笛卡尔积中，其中零的限制和扩展被很好地定义。特别地，在每个子域边界上的诺依曼迹空间被构建为在每个相关联的接口上的分段分布。通过使用内部Calderón投影仪，该问题是投在变分Galerkin形式的算子矩阵，其对角线是由块边界积分算子与子域。基于Lions投影引理在非闭子空间上的推广，我们证明了解的存在唯一性。我们还研究了协调边界元Galerkin离散的渐近拟最优性。在2-D的数值实验证实了该方法的有效性和性能匹配的另一种广泛使用的边界元离散。他们还证明了其顺应性不同类型的预处理。
We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in “subdomains”) that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise-distributions over each associated interface. Through the use of interior Calderón projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions’ projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.