The MFS-SVD method for the laplace equation in three dimensions

Antunes, Pedro; Santos, Vinicius; Serranho, Pedro

http://hdl.handle.net/10400.2/20621

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Authors

Antunes, Pedro

Santos, Vinicius

Serranho, Pedro

Abstract(s)

The method of fundamental solutions (MFS) has been widely used to numerically solve boundary value problems involving linear partial differential equations. This method is easy to implement computationally and can be very effective for smooth domains and boundary conditions. The main drawback of the MFS is the ill-conditioning of the associated matrices, which may deteriorate the method’s accuracy. We present three methods to reduce the ill-conditioning of the classical MFS for the Laplace equation defined in bounded star-shaped domains in 3D. The idea is to expand the MFS basis functions in terms of spherical harmonics, in order to use the reduced QR factorization and singular value decomposition to deal with the ill-conditioning, leading to a better function basis that spans the same approximation space as the classical MFS. The numerical results illustrate that these approaches significantly decrease the ill-conditioning, allowing higher accuracy when compared to the classical MFS.