文档介绍：JOURNAL PUTATIONAL PHYSICS 133, 289–309 (1997)
ARTICLE NO. CP975656
An Analysis of Finite Volume, Finite Element, and Finite Difference
Methods Using Some Concepts from Algebraic Topology
Claudio Mattiussi
CLAMPCO Sistemi .—NIRLAB, AREA Science Park, Padriciano 99, 34012 Trieste, Italy
Received May 23, 1996; revised January 17, 1997
plains why it is expedient to use two distinct and dual
In this paper we apply the ideas of algebraic topology to the discretization grids, it shows how they must be staggered
analysis of the finite volume and finite element methods, illuminat- to achieve optimal performance and proposes a technique
ing the similarity between the discretization strategies adopted by for the construction of high order algorithms ply
the two methods, in the light of a geometric interpretation proposed with the physics of the problem on regular and irregular
for the role played by the weighting functions in finite elements.
We discuss the intrinsic discrete nature of some of the factors ap- grids. A final section devoted to an analysis of the discreti-
pearing in the field equations, underlining the exception repre- zation strategies adopted by the finite difference methods
sented by the constitutive term, the discretization of which is main- (FD) underlines the absence in the classical version of FD
tained as the key issue for numerical methods devoted to field [4] of the distinct geometric flavor of FE and FV, suggesting
problems. We propose a systematic technique to perform this task, how this reflects in the performance of formulas obtained
present a rationale for the adoption of two dual discretization grids
and point out some optimization opportunities in bined with it. New approaches to FD [13, 14] are also briefly
selection of interpolation functions and cell geometry for the finite analyzed mented
volume method. Finally, we suggest an explanation for the intrinsic For concreteness, in the course of the exposition we will
limit