The finite volume method and application in combinations

Abstract

In this paper, the conventional finite volume method (FVM) is interpreted as a new kind of Galerkin finite element method (FEM), where the same piecewise linear functions are chosen as in both trial and test spaces, and some specific integration rules are adopted. Error analysis is made for the regular Delaunay triangulation involving obtuse triangles separated, to prove optimal convergence rates of the approximate solutions obtained. The new interpretation makes the FVM analysis much easier because we may bypass verification of the nontrivial Ladyzhenskaya–Babuska–Brezzi (LBB) condition by the Petrov–Galerkin FEM in the existing analysis of FVM. More importantly, the new interpretation and the simple FVM analysis enable us to construct easily the combinations of FVM with other popular numerical methods, such as the finite element method (FEM), the finite difference method (FDM), the Ritz–Galerkin method (RGM), etc., for solving complicated problems of partial differential equations (PDE). For example, for solving singularity problems, the combination of RGM–FVM is superior to the combination of RGM–FDM in flexibility of arbitrary solution domains, and also superior to the combination of RGM–FEM in substantial saving of CPU time. Since the conservative law of flux may be maintained exactly in the numerical solutions, and since obtuse triangles may be included in the Delaunay triangulation, the FVM and its combinations become very promising for solving elliptic boundary value problems, in particular those where singularity solutions exist and those where the obeying of conservative law is crucial. The numerical examples of combinations of RGM–FVM are given for solving Motz's problems, to verify the optimal convergence rates. The techniques and analysis of FVM and its combinations can be extended to the convection–diffusion problems. Most importantly, an important aspect in this paper is the possibility to include obtuse triangles in the error analysis.