A Novel Extremum-Preserving Finite Element Scheme for Convection Dominant Diffusion Problems

Source abstract

This paper proposes a novel extremum-preserving finite element (FE) scheme for convection-dominated diffusion problems. The main contributions of this paper have four aspects: Firstly, one treatment technique is put forward by ghost ideas for the linear upwind FE scheme applied to such problems with mixed boundary conditions. Secondly, one asymptotic expansion of gradient functions over the support of each node is derived by introducing a linear interpolation extension function on the upwind element. Thirdly, one nonlinear second-order FE discrete operator for the convection term is constructed by the asymptotic expansion and error estimation of its upwind FE solution, where some perturbed coefficients are skillfully introduced into this asymptotic expansion, which serves as a high-order correction. Hence, a novel second-order extremum-preserving FE scheme is designed for convection-dominated diffusion problems by combining the novel discrete convection operator with an appropriate discrete diffusion operator. Fourthly, one efficient local modification algorithm is devised not only to reduce the cost of expensive nonlinear iterations in the above scheme, but also to handle the cases in which the novel scheme is not extremum-preserving. Furthermore, the scheme is extremum-preserving and features universality across spatial dimensions, independence from mesh information, and simplicity in implementation. Numerical results confirm the validity of the theoretical results.