Superconvergence analysis of a two-grid method for semilinear parabolic equations

Abstract In this paper, the superconvergence analysis of a two-grid method (TGM) is established for the semilinear parabolic equations. Based on the combination of the interpolation and Ritz projection technique, an important ingredient in the method, the superclose estimates in the H 1 -norm are deduced for the backward Euler fully-discrete TGM scheme. Moreover, through the interpolated postprocessing approach, the corresponding global superconvergence result is derived. Finally, some numerical results are provided to confirm the theoretical analysis, and also show that the computing cost of the proposed TGM is only half of the conventional Galerkin finite element methods (FEMs).

[1]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[2]  Chuanjun Chen,et al.  Two-grid methods for finite volume element approximations of nonlinear parabolic equations , 2009 .

[3]  Jinchao Xu,et al.  Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem , 1989 .

[4]  Wei Liu,et al.  Two-grid finite volume element methods for semilinear parabolic problems , 2010 .

[5]  Liu Wei,et al.  TWO KINDS OF TWO-GRID ALGORITHMS FOR FINITE DIFFERENCE SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS , 2010 .

[6]  Junping Wang,et al.  Weak Galerkin finite element methods for Parabolic equations , 2012, 1212.3637.

[7]  Dongyang Shi,et al.  Superconvergence analysis for nonlinear parabolic equation with $$EQ_1^\mathrm{{rot}}$$EQ1rot nonconforming finite element , 2018 .

[8]  Vivette Girault,et al.  Two-grid finite-element schemes for the transient Navier-Stokes problem , 2001 .

[9]  Yinnian He,et al.  Two-Level Method Based on Finite Element and Crank-Nicolson Extrapolation for the Time-Dependent Navier-Stokes Equations , 2003, SIAM J. Numer. Anal..

[10]  Dongyang Shi,et al.  Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with EQ1rot\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg , 2016, Journal of Scientific Computing.

[11]  Dongyang Shi,et al.  Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation , 2014, Appl. Math. Lett..

[12]  Dongwoo Sheen,et al.  P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

[13]  Jinchao Xu,et al.  A two-grid discretization scheme for eigenvalue problems , 2001, Math. Comput..

[14]  Xiaoming He,et al.  Immersed finite element methods for parabolic equations with moving interface , 2013 .

[15]  Jinchao Xu,et al.  Two‐grid methods for time‐harmonic Maxwell equations , 2013, Numer. Linear Algebra Appl..

[16]  Q. Lin,et al.  Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation , 2005 .

[17]  Jinchao Xu,et al.  Error estimates on a new nonlinear Galerkin method based on two-grid finite elements , 1995 .

[18]  Luoping Chen,et al.  Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods , 2013 .

[19]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[20]  Yinnian He,et al.  A multilevel finite element method in space‐time for the Navier‐Stokes problem , 2005 .

[21]  Dongyang Shi,et al.  Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation , 2017, Appl. Math. Comput..

[22]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[23]  Yadong Zhang,et al.  Superconvergence of an H1-Galerkin nonconforming mixed finite element method for a parabolic equation , 2013, Comput. Math. Appl..

[24]  Graeme Fairweather,et al.  An H1-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term , 2002, SIAM J. Numer. Anal..