Abstract
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.
Highlights
For an open bounded polygonal domain ⊆ Rd, d = 2, 3, with Lipschitz boundary, we consider in this work the steady-state convection–diffusion– reaction equation
We show that the resulting scheme satisfies the discrete maximum principle (DMP) and give an analysis of the method
Below we present a more detailed study of the behavior of the nonlinear solver with respect to the value of p.We stress the fact that, for any value of p, the function αE is equal to 1 if wh has a local extremum in one of the end points of the edge E. This property is of fundamental importance for the proof of the discrete maximum principle below
Summary
Most shock capturing techniques suffer from the strong nonlinearity introduced when the diffusion coefficient is made to depend on the finite element residual (and the gradient of the approximation function) Because of this the analysis of such methods is incomplete even when linear model problems with constant coefficients are considered. It was shown that the DMP, and even the convergence of the discrete solution to the continuous one, depend on the geometry of the mesh Another approach to combine monotone (low order) finite element methods with linear diffusion and high order FEM using flux-limiters was proposed very recently in [13].
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