An exponential fitting scheme for general convection-diffusion equations on tetrahedral meshes
It extends an existing exponential fitting scheme to general coefficient matrices, providing a stable and monotone method for convection-diffusion equations on simplicial meshes.
The paper constructs and analyzes a finite element scheme for convection-dominated diffusion problems with full coefficient matrices on tetrahedral meshes, achieving first-order convergence under minimal smoothness and monotonicity under certain mesh conditions.
This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in $R^d$, $d=2,3$. This construction is quite close to the scheme of Xu and Zikatanov (Math. Comp. 1999) where a diagonal coefficient matrix has been considered. The scheme is of the class of exponentially fitted methods that does not use upwind or checking the flow direction. It is stable for sufficiently small discretization step-size assuming that the boundary value problem for the convection-diffusion equation is uniquely solvable. Further, it is shown that, under certain conditions on the mesh the scheme is monotone. Convergence of first order is derived under minimal smoothness of the solution.