Convection-diffusion-reaction equations model the conservation of scalar quantities. From analytic point view, solutions these satisfy, under certain conditions, maximum principles, which represent physical bounds solution. That same are respected by numerical approximations solution is often utmost importance in practice. The mathematical formulation this property, contributes to consistency a method, called discrete principle (DMP). In many applications, convection dominates diffusion...

A family of algebraic flux correction (AFC) schemes for linear boundary value problems in any space dimension is studied. These methods' main feature that they limit the fluxes along each one edges triangulation, and we suppose limiters used are symmetric. For an abstract problem, existence a solution, uniqueness solution linearized priori error estimate proved under rather general assumptions on limiters. particular (but standard practice) choice limiters, it shown local discrete maximum...

For the case of approximation convection-diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes scheme satisfy discrete maximum principle. The shown to be Lipschitz and linearity preserving. Using these properties we provide full stability error analysis, which, in dominated regime, shows existence, uniqueness optimal convergence. Then algebraic flux correction method recalled show present can interpreted as...

This work concerns the development of stabilized finite element methods for Stokes problem considering nonstable different (or equal) order velocity and pressure interpolations. The approach is based on enrichment standard polynomial space component with multiscale functions which no longer vanish boundary. On other hand, since test function enriched bubble-like functions, a Petrov--Galerkin employed. We use such strategy to propose stable variational formulations continuous piecewise linear...

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

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 method is using link between AFC nonlinear edge-based diffusion schemes. Then, specific versions method, that is, different definitions limiters, their main stated. Numerical studies compare scheme.

This paper introduces a novel approach to approximate broad range of reaction–convection–diffusion equations using conforming finite element methods while providing discrete solution respecting the physical bounds given by underlying differential equation. The main result this work demonstrates that numerical achieves an accuracy [Formula: see text] in energy norm, where represents polynomial degree. To validate approach, series experiments had been conducted for various problem instances....