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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

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The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important open problems in these fields are discussed. The exposition concentrates on H 1 -conforming finite elements.

Recent results on the numerical analysis of algebraic flux correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.