Abstract
A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.
Highlights
It is well known that convection dominated-diffusion problem (1) often presents serious numerical difficulties
There have appeared many effective discretization schemes concentrating on the hyperbolic nature of the equation, for example, characteristic FD streamline diffusion method [4, 5], Eulerian-Lagrangian method [6, 7], characteristic-finite volume element method [2, 8, 9], Journal of Applied Mathematics characteristics-mixed covolume method [10, 11], the modified method of characteristic-Galerkin FE procedure [12], characteristic nonconforming FEM [13,14,15], characteristic mixed finite element method (MFEM) [16,17,18,19] and expanded characteristic MFEM [1, 20], and so forth
By employing some distinct characters of the interpolation operators on the element instead of the mixed or expanded mixed elliptic projection used in [1, 17, 20] which is an indispensable tool in the traditional characteristic MFEM analysis, the O(h2) order error estimate in L2-norm for original variable u, which is one order higher than [1, 20] and half order higher than [18], is derived, and the optimal error estimates with order O(h) for auxiliary variable σ in L2norm and for u in broken H1-norm are obtained, respectively
Summary
Consider the following convection-dominated diffusion problem: ut + a (x, y) ⋅ ∇u − ∇ ⋅ (b (x, y) ∇u). By employing some distinct characters of the interpolation operators on the element instead of the mixed or expanded mixed elliptic projection used in [1, 17, 20] which is an indispensable tool in the traditional characteristic MFEM analysis, the O(h2) order error estimate in L2-norm for original variable u, which is one order higher than [1, 20] and half order higher than [18], is derived, and the optimal error estimates with order O(h) for auxiliary variable σ in L2norm and for u in broken H1-norm are obtained, respectively It seems that the result for u in broken H1-norm has never been seen in the existing literature by making full use of the high-accuracy estimates of the lowest order Raviart-Thomas element proved by the technique of integral identities in [27] and the special properties of nonconforming EQ1rot element (see Lemma 1 below). Throughout this paper, C denotes a generic positive constant independent of the mesh parameters h and Δt with respect to domain Ω and time t
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
