Approximating second‐order vector differential operators on distorted meshes in two space dimensions

Abstract

SUMMARYA new finite volume method is presented for approximating second-order vector differential operators in two space dimensions. This method allows distorted triangle or quadrilateral meshes to be used without the numerical results being too much altered. The matrices that need to be inverted are symmetric positive definite; therefore, the most powerful linear solvers can be applied. The method has been tested on a few second-order vector partial differential equations coming from elasticity and fluids mechanics areas. These numerical experiments show that it is second-order accurate and locking-free.

Finite volume schemes for diffusion equations: Introduction to and review of modern methods

“…We can however notice that this CeVeFE-DDFV method (as well as the 2D DDFV scheme) is a Gradient Scheme 76 and, therefore, that its convergence without regularity assumptions, for (1.1) as well as nonlinear and nonlocal equations, follows from the general convergence analysis of Gradient Schemes. 55,76 As HMM methods, DDFV methods have been adapted to more complex models than (1.1): nonlinear elliptic equations, 17,24,38 stationary and transient convectiondiffusion equations, 40,93 the cardiac bidomain model, 16 div-curl problems, 44,91 degenerate hyperbolic-parabolic problems 15 (with assumptions on the mesh, see Sec. 5.2), the linear Stokes equations with varying viscosity, 100,101 semiconductor models 33 and the Peaceman model.…”

A finite volume method for approximating 3D diffusion operators on general meshes

“…An other way (that we have chosen) consists in imposing u f = v f and u d = v d for all n f ; x d 2 C d when iterative methods are used for solving the linear system (15).…”

“…The indirect dual mesh lends itself better to the 3D framework and it provides a more accurate approximation of the gradient of the solution. Since then this type of method has been called discrete duality finite volume (DDFV) method in order to emphasize that it satisfies a discrete integration par parts (see [6]) and it proved to be efficient for dealing with several problems arising in various areas of computational physics (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]). Convergence analysis have been carried out in [5,12,18], for the linear case, and in [11] for the broad class of non-linear Leray-Lions type diffusion operators.…”