A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Abstract

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed.The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that thewhere k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.

A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type

“…Compared to standard MFE methods, the EMFE method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux. In recent years, many numerical method based on the expanded mixed scheme, such as two-grid EMFE methods [19][20][21][22], 1 -Galerkin EMFE methods [23][24][25], expanded mixed covolume method [26], expanded mixed hybrid method [27], and positive definite EMFE method [28], have been proposed and discussed.…”

An expanded mixed finite element method for two-dimensional Sobolev equations

“…The expanded mixed finite element(EMFE) method [15,16,17], a new formulation expanding the standard mixed formulation, introduces three (or more) auxiliary variables for practical problems. This method was developed and analyzed by Arbogast, Wheeler and Yotov ( [18]).…”

A New Expanded Mixed Element Method for Convection-Dominated Sobolev Equation

“…From then on, the expanded mixed element method has been applied to solving other partial differential equations [ 10 ]. At the same time, many researchers proposed and studied some new numerical methods based on Chen's expanded mixed method, such as expanded mixed hybrid methods [ 11 ], two-grid expanded mixed finite element method [ 12 – 14 ], expanded characteristic-mixed element method [ 15 ], expanded mixed covolume method [ 16 , 17 ], and expanded positive definite mixed method [ 18 ].…”