The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which potentially might pollute the computed velocity. Mathematically, these methods are not robust in the sense that a contribution from the righthand side which influences only the pressure in the continuous equations possesses an impact on both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergencefree solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods which an appropriate projection of the test function. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained with utilizing robust discretizations.

SUMMARYThis paper presents a numerical study of a two-dimensional time-dependent ow around a cylinder. Its main objective is to provide accurate reference values for the maximal drag and lift coe cient at the cylinder and for the pressure di erence between the front and the back of the cylinder at the ÿnal time. In addition, the accuracy of these values obtained with di erent time stepping schemes and di erent ÿnite element methods is studied.

No abstract

No abstract

No abstract

No abstract

No abstract

Abstract. A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.Key words. algebraic flux correction method, linear boundary value problem, well-posedness, discrete maximum principle, convergence analysis, convection-diffusion-reaction equations.AMS subject classifications. 65N12, 65N301. Introduction. Many processes from nature and industry can be modelled using (systems of) partial differential equations. Usually, these equations cannot be solved analytically. Instead, only numerical approximations can be computed, e.g., by using a finite element method (FEM). The Galerkin FEM replaces just the infinite-dimensional spaces from the variational form of the differential equation with finite-dimensional counterparts. However, if the considered problem contains a wide range of important scales, the Galerkin FEM does not give useful numerical results unless all scales are resolved. For many problems, the resolution of all scales is not affordable because of the huge computational costs (memory, computing time). The remedy consists in modifying the Galerkin FEM in such a way that the effect of small scales is taken into account already on grids which do not resolve all scales. This methodology is usually called stabilization. The most common strategy modifies or enriches the Galerkin FEM, e.g., such that the new discrete problem provides additional control of the error in appropriate norms. An alternative approach acts on the algebraic level, i.e., algebraic representations of discrete operators and vectors are modified before computing a numerical solution. This paper studies a method of the latter type.Applications of algebraically stabilized FEMs can be found in particular for convection-dominated problems. Their construction, e.g., in [18,16,17], is performed for transport equations and they are called flux-corrected transport (FCT) schemes (see also [7] for their application to compressible flows). These schemes can be used also for the discretization of time-dependent convection-diffusion equations, e.g., as in [4,11], where the convection-diffusion equations are part of population balance systems. In [11] it is explicitly emphasized that the FCT scheme was preferred to