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Collocation methods for Poisson's equation

机译:泊松方程的配置方法

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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.
机译:在本文中,我们对搭配方法(CM)进行了分析,该方法使用了大规模的可允许函数,例如正交多项式,三角函数,径向基函数和特定解等。可允许的函数可以分段选择即在不同的子域中使用不同的功能。关键思想是,可以将配置方法视为涉及积分逼近的最小二乘法,并且基于传统的有限元方法分析,可以轻松实现最佳收敛速度。关键分析是证明一致的V_h-椭圆不等式和所使用的一些逆不等式。本文探讨了一个有趣的事实,即对于本文给出的配置方法,积分规则仅影响均匀V_h-椭圆不等式,而不影响求解精度。 CM的优点是可以轻松地建立关联的代数方程,可以通过最小二乘法直接从搭配方程中求解,从而大大减少了关联矩阵的条件数。此外,提出了新的有效条件编号,以提供更好的条件编号上限,并为通过搭配方法解决的实际问题显示出良好的稳定性。注意Li [Z.C. Li,具有奇异性,界面和无穷大的椭圆方程的组合方法,Kluwer Academic Publishers,波士顿,伦敦,1998年]是CM的特例,其中允许函数完全满足方程。还对泊松问题进行了数值实验,以支持所做的分析。

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