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Meshless method for numerical solution of PDE using Hermitian interpolation with radial basis. (English) Zbl 1082.65128

Chen, Falai (ed.) et al., Geometric computation. River Edge, NJ: World Scientific (ISBN 981-238-799-4/hbk). Lecture Notes Series on Computing 11, 209-220 (2004).
Summary: In computer aided geometric design, many curves and surfaces, which we want to design, are the solutions of some partial differential equations. The corresponding discrete equations often appear as Hermite-Birkhoff interpolations. Such equations are very difficult to solve in the function space of piecewise polynomials on mesh, since the solutions of the partial differential equations are very smooth even in \(C^\infty\).
The authors discuss a meshless method for numerical solution of partial differential equations (PDEs) by using the Hermite-Birkhoff interpolation with radial basis, which is generalized from the thin plate spline. This method is a direct discretion of collocation type for ordinary and partial differential equations, with possibility of generalization to integral equations or even equations with time delays.
If we adopt the results in the discussion of radial basis and the Hermite-Birkhoff interpolation, the solvability of the discrete system of equations can be proven on very weak assumptions. The order of the approximation of the scheme depends on the smoothness of the solution and the order of the differential equation.
For the entire collection see [Zbl 1073.65004].

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35G15 Boundary value problems for linear higher-order PDEs
65D05 Numerical interpolation
65D17 Computer-aided design (modeling of curves and surfaces)
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