期刊
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
卷 27, 期 3, 页码 554-580出版社
WILEY
DOI: 10.1002/num.20539
关键词
collocation method; convergence; inverse inequality; reproducing kernel approximation; stability
资金
- National Science Council (Taiwan, Republic of China)
- Lawrence Livermore National Laboratory (USA)
Solving partial differential equations (PDE) with strong form collocation and nonlocal approximation functions such as orthogonal polynomials, trigonometric functions, and radial basis functions exhibits exponential convergence rates; however, it yields a full matrix and suffers from ill conditioning. In this work, we discuss a reproducing kernel collocation method, where the reproducing kernel (RK) shape functions with compact support are used as approximation functions. This approach offers algebraic convergence rate, but the method is stable like the finite element method. We provide mathematical results consisting of the optimal error estimation, upper bound of condition number, and the desirable relationship between the number of nodal points and the number of collocation points. We show that using RK shape function for collocation of strong form, the degree of polynomial basis functions has to be larger than one for convergence, which is different from the condition for weak formulation. Numerical results are also presented to validate the theoretical analysis. (c) 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 554-580, 2011
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