H¹‐Galerkin expanded mixed finite element methods for nonlinear pseudo‐parabolic integro‐differential equations

Abstract

H1‐Galerkin mixed finite element method combined with expanded mixed element method is discussed for nonlinear pseudo‐parabolic integro‐differential equations. We conduct theoretical analysis to study the existence and uniqueness of numerical solutions to the discrete scheme. A priori error estimates are derived for the unknown function, gradient function, and flux. Numerical example is presented to illustrate the effectiveness of the proposed scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

An $$H^1$$ H 1 -Galerkin mixed finite element method for time fractional reaction–diffusion equation

“…Now, we estimate the last term on the right hand side of (37). Using the similar result to (26), we have…”

“…At the same time, we are trying to find some new discrete methods for approximating fractional derivatives and study some other MFE procedures [37,42] based on moving finite element method [1] for solving the fractional PDEs.…”

“…Compared to classical mixed methods, this method has several distinct characteristics: First, it is free of the LBB consistency condition; Second, the polynomial degrees of the finite element spaces V h and W h may be different; Third, the optimal H 1 -error estimates for both the scalar unknown u and its gradient σ are obtained. In view of the method's attractive features, the one has been used to seek the numerical solutions of some integer order partial differential equations [28][29][30][31][32][33][34][35][36][37][38][39]. However the numerical analysis of H 1 -Galerkin MFE method of fractional PDEs has not been studied and discussed.…”

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

“…[39], to nonlinear pseudo-parabolic integro-differential equations in Ref. [40] and to a nonlinear parabolic equation in porous medium flow in Ref. [41].…”

Weak Galerkin finite element methods for linear parabolic integro-differential equations

“…For earlier work on numerical solutions of parabolic integro-differential equations, see for example, [1][2][3][4][5][6][7][8][9]. In recent years, several finite element/volume techniques for equations above have been studied, including mixed finite element methods [10][11][12], expanded mixed finite element methods [13,14], expanded mixed covolume methods [15], h p -local discontinous Galerkin methods [16], discontinuous mixed covolume methods [17], and least-squares finite element methods [18].…”