Abstract
A meshless technique is presented based on a special scattered data interpolation method which converts the original problem to a higher order differential equation, typically to an iterated Laplace or Helmholtz equation. The conditions of the original problem (interpolation conditions, boundary conditions and also the differential equation) are taken into account as special, non-usual boundary conditions taken on a finite set of collocation points. For the new problem, existence and uniqueness theorems are proved based on variational principles. Approximation properties are also analysed in Sobolev spaces. To solve the resulting higher order differential equation, robust quadtree/octtree-based multi-level techniques are used, which do not need any spatial and/or boundary discretisation and are completely independent of the original problem and its domain. This approach ca be considered as a special version of the method of radial basis functions (based on the fundamental solution of the applied differential operator) but avoids the solution of large and poorly conditioned systems, which significantly reduces the memory requirements and the computational cost as well.
This research was partly sponsored by the Hungarian Scientific Research Fund (OTKA) under the project T34652.
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References
Floater, M.S., Iske, A.: Multistep Scattered Data Interpolation Using Compactly Supported Radial Basis Functions. Journal of Computational and Applied Mathematics, 1996, 73, 65–78.
Franke, R.: Scattered Data Interpolation: Test of Some Methods. Mathematics of Computation, 1982, 38 (157), 181–200.
Gáspár, C., Simbierowicz, P.: Scattered Data Interpolation Using Unstructured Grids. In: HYDROCOMP ’92 (ed. by J. Gayer, Ö. Starosolszky, C. Maksimovic), pp. 131–138. Proceedings of the Int. Conf. on Comput. Methods and Measurements in Hydraulics and Hydrology, Budapest, Hungary, 1992. Water Resources Research Centre, Budapest, 1992.
Gáspár, C: Flow Modelling Using Quadtrees and Multigrid Techniques. Proceedings of the Third International Conference on Computational Structures Technology, Budapest, Hungary, 21–23 August, 1996. Civil-Comp Press, 1996.
Gáspár, C: Biharmonic and bi-Helmholtz Type Scattered Data Interpolation Using Quadtrees and Multigrid Technique. In: Multigrid Methods VI. Proceedings of the Sixth European Multigrid Conference, Gent, Belgium, 27–30 September, 1999. (Ed. by E.Dick, K. Riemslagh, J. Vierendeels.) Springer, 2000.
Gáspár, C: Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method. Engineering Analysis with Boundary Elements, 2000, 24 (7–8), 559–573.
Golberg, M.A., Chen, C.S.: The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Boundary Element Communications, 1994, 5 (2), 57–61.
Golberg, M.A., Chen, C.S.: A bibliography on radial basis function approximation, Boundary Element Communications, 1996, 7 (4), 155–163.
Golberg, M.A., Chen, C.S., Bowman, H.: Some recent results and proposals for the use of radial basis functions in the BEM Engineering Analysis with Boundary Elements, 23 (1999), 285–296.
Hardy, R.L.: Theory and Applications of the Multiquadric-Biharmonic Method, 20 Years of Discovery, 1968–1988. Computers Math. Applic. 1990, 19 (8/9), 163–208.
Kansa, E.J.: Multiquadrics — a Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics I.–II. Computers Math. Applic. 1990, 19 (8/9), 127–161.
Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4 (1995), 389–396.
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Gáspár, C. (2003). Fast Multi-Level Meshless Methods Based on the Implicit Use of Radial Basis Functions. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56103-0_11
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DOI: https://doi.org/10.1007/978-3-642-56103-0_11
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