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A local radial basis function method for the Laplace-Beltrami operator. (English) Zbl 1475.65141

This article discusses a local meshfree method for the approximation of the Laplace-Beltrami operator on a smooth surface in 3D. The approach relies on radial basis functions augmented with multivariate polynomials. Also, the approach does not require an explicit expression of the surface, which can be simply defined by a set of scattered nodes. The convergence, accuracy and other computational characteristics of the proposed method are discussed. Numerical experiments related to the Turing model for pattern formation and the Schaeffer’s model for electrical cardiac tissue behavior are included.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65D12 Numerical radial basis function approximation
65D05 Numerical interpolation
35R01 PDEs on manifolds
58J05 Elliptic equations on manifolds, general theory
35K57 Reaction-diffusion equations
92C15 Developmental biology, pattern formation
92B25 Biological rhythms and synchronization

Software:

SG; rbf_qr; Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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