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

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.

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.

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov-Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods. Introduction.Finite element solution of the Stokes problem poses the basic problem of satisfying the discrete Babuska-Brezzi (or inf-sup) condition (see [24] and the references therein). This is indeed a restriction from the point of view of implementation since equal order velocity and pressure spaces do not satisfy this condition. On the other hand, the minimal space to imagine, namely continuous piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure, does not satisfy this condition either.Several solutions have been proposed to overcome this restriction, starting with that in [11] and the first consistent method in [28]. Moreover,in [23,27,29,34] the possibility of considering discontinuous spaces for the pressure was considered and justified. On the other hand, in [14,13], the idea from [16] has been used to propose a new kind of stabilized finite element methods, with stabilizing terms now containing only jump terms across the interelement boundaries. For an overview of stabilized finite element methods for the Stokes problem, see [19] and [5].On the other hand, the theoretical justification of stabilized methods has become a subject of interest in the last decade. In [2,3,4,31], the connection between stabilized finite element methods and Galerkin methods enriched with bubble functions has been used to propose new stabilized finite element methods for Stokes-like and linearized Navier-Stokes problems. Also, in [22] macro bubbles were used to derive a method analogous to the locally stabilized method from [29] containing jump terms across the interelement boundaries of the macroelements. In the resulting method, the stabilizing

This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection–diffusion problems confirm the theory.

SUMMARYFully computable upper bounds are developed for the discretisation error measured in the natural (energy) norm for convection–reaction–diffusion problems in three dimensions. The upper bounds are genuine upper bounds in the sense that the numerical value of the estimated error exceeds the actual numerical value of the true error regardless of the coarseness of the mesh or the nature of the data for the problem. All constants appearing in the bounds are fully specified. Examples show the estimator to be reliable and accurate even in the case of complicated three‐dimensional problems. Copyright © 2013 John Wiley & Sons, Ltd.

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