Unified convergence analysis of numerical schemes for a miscible displacement problem

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $L^\infty(0,T; L^2(\Omega))$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion

An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes

International audienceWe are interested here in the numerical approximation of a family of probability measures, solution of the Chapman-Kolmogorov equation associated to some non-diffusion Markov process with Uncountable state space. Such an equation contains a transport term and another term, which implies redistribution Of the probability mass on the whole space. All implicit finite Volume scheme is proposed, which is intermediate between an upstream weighting scheme and a modified Lax-Friedrichs one. Due to the seemingly unusual probability framework, a new weak bounded variation inequality had to be developed, in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as for the study of finite Volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in $L^1$. Also, due to the redistribution term, the tightness of the family of approximate probability measures had to be proven. Numerical examples are provided, showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context

Orlicz capacities and applications to some existence questions for elliptic pdes having measure data

We study the sequence un, which is solution of $-{\rm div}(a(x,{\nabla}u_n)) + \Phi''(|u_n|)\,u_n= f_n+ g_n$ in Ω an open bounded set of RN and un= 0 on ∂Ω, when fn tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N-function Φ, and prove a non-existence result

Stability of renormalized solutions of elliptic equations with measure data

We prove the stability of Dirichlet problems of the type div (a " (x; u " ; ru " )) = " ; u " = 0; with respect to suitable convergences of the nonlinear operators a " and of the measure data " . As a consequence, we obtain the existence of a renormalized solution for a general class of nonlinear elliptic equations with right-hand side measure

Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data

Let u(n) be the sequence of solutions of -div(a(x, u(n), delu(n)))+u(n)(q-1) u(n) = f(n), in Omega, u(n) = 0 on deltaOmega, where Omega is a bounded set in R-N and f(n) is a sequence of functions which is strongly convergent to a function f in L-loc(1)(OmegaK), with K a compact in Omega of zero r-capacity, no assumptions are made on the sequence f. on the set K. We prove that if a has growth of order p-1 with respect to delu (p > 1), and if q > r(p-1)/(r -p), then u(n) converges to u, the solution of the same problem with datum f, thus extending to the nonlinear case a well-known result by H. Brezis. (C) 2002 Elsevier Science (USA)