差分格式与有限元格式的某些统一性

本文讨论了在矩形网格剖分情况下求解Poisson方程的差分方法与有限元方法的某些统一性,由此可知差分格式和有限元格式之间存在某种线性组合关系,差分解也是弱解。这些结论使数值计算得到进一步简化。

Identity Features of Difference Scheme and Finite Element Scheme on Rectangular Grid

The difference scheme and finite element scheme on a rectangular grid for solving Poisson's equation are discussed. Some identity features of two discrete schemes have been noticed. In particular, a certain linear combination relation exists between them. Three important conclusions have been formed: (a) The difference solution for an elliptical boundary problem is a weak one, and the numerical results obtained either with a difference scheme or with a finite element scheme will be practically identical if the smoothness of the solution is not good enough; (b) The discrete scheme of higher accuracy can be constructed by a combination of lower accuracy; (c) The identy features are useful for the study of parallel and splitting algorithms and very convenient in practical computation.