Finite Volume Methods for Diiusion Convection Equations on General Meshes
abstract Error estimates for the approximate solution of stationary or evo-lutive convection diiusion equations using a nite volume scheme on a general non structured mesh are presented here. These estimates only require a piece-wise regularity of the exact solution.
Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes
A discretization of heterogeneous and anisotropic diffusion problems on general discretization meshes is developed and studied. The unknowns of this scheme are the values at the center of the control volumes and at some internal interfaces, chosen because of some irregularity of the diffusion tensor. If the tensor is regular enough, the values on the interfaces may be deduced from the values at the center, at the expense of loosing the local conservativity of the fluxes. This scheme is shown to be accurate on several numerical examples. Mathematical convergence to the continuous solution is obtained for homogeneous and heterogeneous tensors. An error estimate may be drawn under sufficient regularity assumptions on the solution.
A uniqueness result for measure-valued solutions of nonlinear hyperbolic equations
We study here the solutions of the nonlinear hyperbolic equation ut + div(vf (u)) = 0 in IR N 0; T ], with given initial condition u(:; 0) = u0(:) in IR N , where v is a given function from IR N 0; T ] to IR N and f is a given function from IR to IR. We prove that if is a measure valued solution satisfying some entropy condition, which we deene, then x;t = u(x;t) for a.e. (x; t) 2 IR N 0; T ], where u is the unique entropy weak solution to the equation. We consider here the following nonlinear hyperbolic equation, with initial condition: (u t (x; t) + div(vf(u(x; t))) = 0; x 2 IR N ; t 2 0; T] u(x; 0) = u 0 (x); x 2 IR N (1) where T > 0, u t denotes the derivative of u with respect to t, divv= N X i=1 N). It is assumed that there exists V > 0 such that supfjv(x; t)j; x 2 IR N ; t 2 0; T]g V , where j:j denotes the Euclidean norm in IR N , that f is a given function of class C 3 from IR to IR, and that u 0 2 L 1 (IR N) is a given function. It is also assumed that supf?div(v(x; t))f 0 (u), x 2 IR N , t 2 0; T], u 2 IRg < 1 (this is true, for instance, if divv = 0). Under these assumptions, Kruzkov 4] gives results of existence (and uniqueness) of the entropy weak solution u, i.e. a function u 2 L 1 (IR N ]0; T) which satisses : Z IR N Z T 0 (u(x; t))' t (x; t) + (u(x; t))v(x; t):grad'(x; t))dtdx + Z IR N Z T 0 divv(x; t)'(x; t) (u(x; t)) ? 0 (u(x; t)f(u(x; t)) dtdx+ Z IR N