โ Scribed by Mikhail Shashkov; Stanly Steinberg
No coin nor oath required. For personal study only.
An algorithm is developed for discretizing boundary-value problems given by a general linear elliptic second-order partial-differential equation with general mixed or Robin boundary conditions in general logically rectangular grids. The continuum problem can be written as an operator equation where the operator is self adjoint and positive definite. The discrete approximations have the same property. Consequently, the matrices for the discrete problem are symmetric and positive definite. Also, the scheme has a nearest neighbor stencil. Consequently, the most powerful linear solvers can be applied. In smooth grids, the algorithm produces secondorder accurate solutions. It is the generality of the problem (general matrix coefficients, general boundary conditions, general logically rectangular grids) that makes finding such an algorithm difficult. The algorithm, which is a combination of the method of support operators and the mapping method, overcomes certain difficulties of the individual methods, producing a high-quality algorithm for solving general elliptic problems. 1995 Academic Press, Inc.
๐ SIMILAR VOLUMES
After it is shown that the classical five-point mesh-centered finite difference scheme can be derived from a low-order nodal finite element scheme by using nonstandard quadrature formulae, higher-order block mesh-centered finite difference schemes for second-order elliptic problems are derived from
## Abstract We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal __H__^1^ error estimate, __H__^1^ superconvergence and __L__^__p__^ (1 < __p__ โค โ) error estimates betw