Renovated Finite Volume Methods for Anisotropic Diffusion Problems Z. C. Li, B. M. Holec, L. J. Young and M. G. Lee ∗ , May 28, 2011 Abstract The finite volume method (FVM) was studied in Li and Wang [7] with the error analysis in H 1 norm, and is combined with other methods, such as the finite element method (FEM) and the collocation Trefftz method (CTM) [11]. In this paper, the renovated FVM is developed for the anisotropic diffusion problem, where there exist the mixed derivatives ∂ 2 u ∂x∂y . The computational algorithms are provided, and new partitions are proposed for existing obtuse triangles. The new FVM in this paper is as flexible as FEM, but it has two remarkable advantages over FEM: (1) The discrete algebraic equations can be formulated simple and straightforward; (2) The conservative law of flux is always obeyed for the anisotropic diffusion problem. Hence, the renovated FVM may find wide application in engineering problems. 1 Introduction In this note, consider the anisotropic diffusion problem [5]: Find u ∈ C 2 (S ) such that Lu = −{ ∂ ∂x (K 11 ∂u ∂x + K 12 ∂u ∂y )+ ∂ ∂y (K 21 ∂u ∂x + K 22 ∂u ∂y )} + cu = f in S, (1.1) u = g 1 , on Γ D , (1.2) (K 11 ∂u ∂x + K 12 ∂u ∂y ) cos(ν, x)+(K 21 ∂u ∂x + K 22 ∂u ∂y ) cos(ν, y) (1.3) +αu = g 2 , on Γ M , * Corresponding author 1