A new method of fundamental solutions applied to nonhomogeneous elliptic problems
A new method of fundamental solutions applied to nonhomogeneous elliptic problems
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DOI：
10.1007/s10444-004-1833-5
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发表时间：
2005-07
影响因子：
1.7
通讯作者：
Carlos J. S. Alves;Ching-Shyang Chen
中科院分区：
文献类型：
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作者：
Carlos J. S. Alves;Ching-Shyang Chen
经典的基本解方法（MFS）仅用于求解齐次偏微分方程。与其他数值方法如区域积分、双倒易方法（多项式或径向基函数插值）相结合，MFS可以扩展到求解非齐次问题。本文提出了一个扩展的MFS的直接逼近泊松和非齐次亥姆霍兹问题。这可以通过使用相关联的本征值方程的基本解作为基础来近似非齐次项来完成。然后可以评估PDE的特定解。这种无网格方法的一个优点是，齐次和非齐次方程的分辨率可以以一种统一的方式结合起来，它可以用于多尺度问题。数值模拟，并显示几个测试示例的近似质量。
The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples.