A new approach to overcome the ill-conditioning of the Method of Fundamental Solutions (MFS) combining Singular Value Decomposition (SVD) and an adequate change of basis was introduced in as MFS-SVD. The original formulation considered polar coordinates and harmonic polynomials as basis functions and is restricted to the Laplace equation in 2D. In this work, we start by adapting the approach to the Helmholtz equation in 2D and later extending it to elliptic coordinates. As in the Laplace case, the approach in polar coordinates has very good numerical results both in terms of conditioning and accuracy for domains close to a disk but does not perform so well for other domains, such as an eccentric ellipse. We therefore consider the MFS-SVD approach in elliptic coordinates with Mathieu functions as basis functions for the latter. We illustrate the feasibility of the approach by numerical examples in both cases.

中文翻译：

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改进极坐标或椭圆坐标域上亥姆霍兹方程基本解法的条件


MFS-SVD 引入了一种新方法来克服基本解法 (MFS) 的病态，该方法结合了奇异值分解 (SVD) 和基础的适当变化。原始公式将极坐标和调和多项式视为基函数，并且仅限于二维拉普拉斯方程。在这项工作中，我们首先将该方法应用于二维亥姆霍兹方程，然后将其扩展到椭圆坐标。与拉普拉斯情况一样，极坐标方法在接近圆盘的域的条件和精度方面都具有非常好的数值结果，但对于其他域（例如偏心椭圆）则表现不佳。因此，我们考虑椭圆坐标中的 MFS-SVD 方法，并以 Mathieu 函数作为后者的基函数。我们通过两种情况下的数值例子说明了该方法的可行性。