The fast multipole method for the symmetric boundary integral formulation
The fast multipole method for the symmetric boundary integral formulation
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DOI：
10.1093/imanum/dri033
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发表时间：
2006-04
影响因子：
2.1
通讯作者：
O. Steinbach;W. Wendland
中科院分区：
文献类型：
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作者：
O. Steinbach;W. Wendland
采用对称伽辽金边元法求解Dirichlet型和Neumann型混合边界条件的边值问题。作为一个模型问题，我们考虑拉普拉斯方程。当采用迭代格式求解得到的线性系统时，采用快速多极方法实现离散边界积分算子。虽然单层势可以像原始算法那样直接实现，但双层势及其伴随算子是通过对单层势核的多极级数应用法向导数来近似的。通过分部积分将超奇异积分算子的伽辽金离散化简化为单层势。最后，我们用边界积分算子的快速多极方法给出了这些近似的稳定性和误差分析。结果表明，快速多极方法的使用并不影响最优渐近收敛性。所得到的线性系统由GMRES方案求解，该方案采用快速多极方法中已经采用的分层策略作为先决条件。我们的数值算例与理论结果一致。
A symmetric Galerkin boundary-element method is used for the solution of boundary-value problems with mixed boundary conditions of Dirichlet and Neumann type. As a model problem we consider the Laplace equation. When an iterative scheme is employed for solving the resulting linear system, the discrete boundary integral operators are realized by the fast multipole method. While the single-layer potential can be implemented straightforwardly as in the original algorithm for particle simulation, the double-layer potential and its adjoint operator are approximated by the application of normal derivatives to the multipole series for the kernel of the single-layer potential. The Galerkin discretization of the hypersingular integral operator is reduced to the single-layer potential via integration by parts. We finally present a corresponding stability and error analysis for these approximations by the fast multipole method of the boundary integral operators. It is shown that the use of the fast multipole method does not harm the optimal asymptotic convergence. The resulting linear system is solved by a GMRES scheme which is preconditioned by the use of hierarchical strategies as already employed in the fast multipole method. Our numerical examples are in agreement with the theoretical results.