Group of Mathematical Physics of the University of Lisbon

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A well conditioned method of fundamental solutions for laplace equation

Publication . Antunes, Pedro R. S.

The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved typically are ill-conditioned and this may prevent to achieve high accuracy. In this work, we propose a new algorithm to remove the ill conditioning of the classical MFS in the context of Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization we define well conditioned basis functions spanning the same functional space as the MFS's. Several numerical examples show that this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.

2022Preprint

Open access

Improving the conditioning of the method of fundamental solutions for the Helmholtz equation on domains in polar or elliptic coordinates

Publication . Antunes, Pedro R. S.; Calunga, Hernani; Serranho, Pedro

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 [1] 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.

2024Journal article

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