Collocation methods for Poisson's equation

摘要

In this paper, we provide an analysis on the collocation methods (CM), which uses a large scale of admissible functions such as orthogonal polynomials, trigonometric functions, radial basis functions and particular solutions, etc. The admissible functions can be chosen to be piecewise, i.e., different functions are used in different subdomains. The key idea is that the collocation method can be regarded as the least squares method involving integration approximation, and optimal convergence rates can be easily achieved based on the traditional analysis of the finite element method. The key analysis is to prove the uniformly V_h-elliptic inequality and some inverse inequalities used. This paper explores the interesting fact that for the collocation methods given in this paper, the integration rules only affect on the uniformly V_h-elliptic inequality, but not on the solution accuracy. The advantage of the CM is to formulate easily the associated algebraic equations, which can be solved from the collocation equations directly by the least squares method, thus to greatly reduce the condition number of the associated matrix. Moreover, the new effective condition number is proposed to provide a better upper bound of condition number, and to show a good stability for real problems solved by the collocation methods. Note that the boundary approximation method in Li [Z.C. Li, Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, Kluwer Academic Publishers, Boston, London, 1998] is a special case of the CM, where the admissible functions satisfy the equations exactly. Numerical experiments are also carried for Poisson's problem to support the analysis made.