For the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.
The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations. Abstract. We consider fully discrete schemes for linear parabolic problems discretized by the CrankNicolson method in time and the standard finite element method in space. We study the effect of mesh modification on the stability of fully discrete approximations as well as its influence on residual-based a posteriori error estimators. We focus mainly on the qualitative, analytical and computational behavior of the schemes and the error estimators.
Abstract. In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection-diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.
Abstract. In this paper we derive a posteriori error estimates for space discrete approximations of the time dependent Stokes equations. By using an appropriate Stokes reconstruction operator we are able to write an auxiliary error equation in pointwise form that satisfies the exact divergence free condition. Thus standard energy estimates from pde theory can be applied directly to yield a posteriori estimates that rely on available corresponding estimates of the stationary Stokes equation.Estimates of optimal order in L ∞ (L 2 ) and in L ∞ (H 1 ) for the velocity are derived for finite element and finite volume approximations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.