Abstract
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.
Highlights
Scalar convection–diffusion equations model the convective and molecular transport of a quantity like temperature or concentration
In contrast to spurious oscillations at layers diminishing (SOLD) methods, which are based on variational formulations, the main idea of algebraic flux correction (AFC) schemes consists in modifying the algebraic system corresponding to a discrete problem, typically the Galerkin discretization, by means of solution-dependent flux corrections
The results of [10,11] indicate that a better convergence behaviour in the diffusion-dominated case may be expected if the AFC scheme is linearity preserving, i.e., if the stabilization originating from the AFC vanishes in regions where the approximate solution is a polynomial of degree 1
Summary
Scalar convection–diffusion equations model the convective and molecular transport of a quantity like temperature or concentration. In contrast to SOLD methods, which are based on variational formulations, the main idea of AFC schemes consists in modifying the algebraic system corresponding to a discrete problem, typically the Galerkin discretization, by means of solution-dependent flux corrections. This study is complemented by the work [10], where a link between the AFC schemes and a nonlinear edge-based diffusion scheme is presented, and the linearity preservation of the scheme is studied in detail. This latter reformulation offers the applicability of different tools than those used so far for the analysis of AFC schemes.
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