An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology

plains why it is expedient to use two distinct and dual discretization grids, it shows how they must be staggered In this paper we apply the ideas of algebraic topology to the analysis of the finite volume and finite element methods, illuminatto achieve optimal performance and proposes a technique ing the similarity between the discretization strategies adopted by for the construction of high order algorithms which comply 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. grids. A final section devoted to an analysis of the discretiWe discuss the intrinsic discrete nature of some of the factors apzation strategies adopted by the finite difference methods pearing in the field equations, underlining the exception represented 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 with it. New approaches to FD [13, 14] are also briefly and point out some optimization opportunities in the combined analyzed and commented selection of interpolation functions and cell geometry for the finite volume method. Finally, we suggest an explanation for the intrinsic For concreteness, in the course of the exposition we will limitations of the classical finite difference method in the construcalmost always refer to a steady heat conduction model tion of accurate high order formulas for field problems. Q 1997 Acaproblem. The choice of thermostatics was made in order demic Press to present the results in a context which, for its simplicity, should be familiar to the widest possible audience. Nonetheless, the discussion applies to every field theory whose