Stable evaluation of gaussian kernel applied to interface problems
Authors:
- Artur Krowiak,
- Jordan Podgórski
Abstract
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.
- Record ID
- CUT5d22aa0c3da94432afc3805569e88a31
- Publication categories
- ; ;
- Author
- Pages
- 85-96
- Other elements of collation
- rys.; Bibliografia (na s.) - 95-96; Bibliografia (liczba pozycji) - 16; Oznaczenie streszczenia - Abstr.
- Book
- Loja Maria Amélia, Maria Amélia Loja Barbosa Joaquim Infante, Joaquim Infante Barbosa Rodrigues José Alberto José Alberto Rodrigues [et al.] (eds.): 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, 2019, Porto, ECCOMAS, ISBN 978-989-99410-5-2
- Keywords in English
- positive definite kernels, Mercer’s series, stable RBF basis, interface problem
- Language
- eng (en) English
- Score (nominal)
- 5
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUT5d22aa0c3da94432afc3805569e88a31/
- URN
urn:pkr-prod:CUT5d22aa0c3da94432afc3805569e88a31
* presented citation count is obtained through Internet information analysis, and it is close to the number calculated by the Publish or PerishOpening in a new tab system.