A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Abstract

This paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t + ∇ · (uf (c)) − ε∆c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1 -norm in the situation, where the diffusion parameter ε is smaller or comparable to the mesh size. Numerical experiments underline the theoretical results. Subject Classification (1991): 65M15, 35K65, 76M25 Mathematics

Reduced basis method for finite volume approximations of parametrized linear evolution equations

“…For (iv) it is sufficient to note, that the explicit components have discretely zero divergence and we are using a monotone convective flux. So with the given CFL condition on the time-step size, application of [28], Lemma 3, yields the coercivity ofL k E , which in particular implies its psd-ness. Note that only (i) and (ii) will be required for well-definedness of the RB-scheme.…”

“…This is clearly satisfied by setting λ := 1/V , where V was the upper bound on the norm of the velocity. Additionally, a CFL-condition for the time-step size must be satisfied such that the explicit evolution-operator is monotone [28]. If diffusive components were discretized explicitly this would lead to the following very restrictive condition ∆t…”

“…If the latter two points are not satisfied by a given scheme, the RB-approach still is applicable, although with rougher error-predictions. The psd-ness of the space-discretization operators is also valid for upwind fluxes as long as the velocity field is (discretely) divergence free [28]. In the nonzero-divergence case, convective upwind fluxes can lead to space-discretization operators, which then are indeed non-coercive.…”

A posteriorierror estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

“…In this paper we continue our work on a posteriori error estimates for finite volume approximations of nonlinear conservation laws and convection diffusion equations, which was started in [33] and [41]. In [41] we analyzed an explicit cell centered finite volume approximation to an unstationary convection diffusion equation and proved an a posteriori error estimator of order (ε 2 + h + ∆t) 1/4 , where ε denoted the small diffusion parameter (e.g.…”

A Posteriori Error Estimates for Finite Volume Approximations

“…E-mail: snicaise@univ-valenciennes.fr direction. See [12,21,1,10,11,18,19,30] for cell centered finite volume methods, [15,16,22] for vertex-centered methods, and [5,13,14] for finite volume element methods.…”