On some RBF collocation methods in the application to problems with discontinuity / Jordan Podgórski, Artur Krowiak // W: SYMCOMP 2019 - 4th International Conference on Numerical and Symbolic Computation: ... więcej

On some RBF collocation methods in the application to problems with discontinuity / Jordan Podgórski, Artur Krowiak // W: SYMCOMP 2019 - 4th International Conference on Numerical and Symbolic Computation: Developments and Applications : ECCOMAS Thematic Conference, ISEP - Instituto Superior de Engenharia do Porto, Portugal, April, 11-12 : proceedings [Dokument elektroniczny] / eds. Maria Amélia Loja [et al.] ; Associação Portuguesa de Mecânica Teórica. Aplicada e Computacional. – Porto : ECCOMAS, 2019. – Pamięć USB. – S. 97-107 : rys., tab. – Bibliogr. 15 poz., Abstr. – ISBN 978-989-99410-5-2

The paper deals with the application of some meshless collocation techniques to problems possessing discontinuity on the interface. The presented methods fall into the category of analytic, approximate ... więcej

The paper deals with the application of some meshless collocation techniques to problems possessing discontinuity on the interface. The presented methods fall into the category of analytic, approximate approaches. They take advantage of radial basis functions (RBFs) to form the approximate, sought solution of a differential equation and by collocating it at chosen points in the domain as well as on the boundary reduce the differential problem to algebraic one. The methods considered in the present paper differ to each other by choosing the collocation points. One of them uses centers of RBFs as collocation points (Kansa approach) leading to square system matrix, the other places collocation points at different positions to RBFs centers and can generate square or overdetermined systems of algebraic equations. In the present paper these two approaches are adapted to the problems possessing a kind of discontinuity on the interface. To this end, the so-called subdomain technique is involved. The application of the methods to such problems has several advantages: their meshless character allows to easily solve the problems with irregular domains as well as irregular interfaces, the use of RBFs allows to easily treat the problems in higher dimensions and global feature of the methods leads to high efficiency (few nodes are sufficient to obtain very accurate results). On the other hand special attention should be paid to the problem of computational stability since it is well known that the global collocation with RBFs can produce highly ill-conditioned system matrices. In the present paper several aspects, such as: accuracy, convergence and stability, of the use of these methods in such problems are studied and compared.