Superconvergence Analysis of an

H^1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^1$$\end{document}

-Galerkin Mixed FEM for Nonlinear Kirchhoff-Type Equation with Damping Term

Abstract

In this paper, a second-order backward differentiation formula (BDF2)

H^1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^1$$\end{document}

-Galerkin mixed finite element method (FEM) is developed for solving the nonlinear Kirchhoff-type equation with a damping term. By introducing a new variable

\mathit{q}=\mathrm{\nabla }{u}_t{+(1+\Vert \mathrm{\nabla }u\Vert }^2)\mathrm{\nabla }u

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{{q}}}=\nabla u_t+(1+\Vert \nabla u\Vert ^2)\nabla u$$\end{document}

, the original hyperbolic equation is transformed into two novel parabolic equations. By means of mathematical induction, the derivative transfer technique, and the technique of recombination for some terms, the superconvergence results with

O(h^2+{\tau }^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(h^2+\tau ^2)$$\end{document}

of u in the

H^1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^1$$\end{document}

-norm and

\mathrm{\nabla }·\mathit{q}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla \cdot {\varvec{{q}}}$$\end{document}

in the

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

-norm are derived in detail. At last, a numerical example is carried out to illustrate the correctness of the theoretical analysis.

Keywords

Nonlinear Kirchhoff-type equation / Damping term /

H^1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^1$$\end{document}

-Galerkin mixed finite element method (FEM) / Second-order backward differentiation formula (BDF2) scheme / Superconvergent results / 35Q55 / 65M60 / 65M22

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Funding

Doctoral Starting Foundation of Pingdingshan University(PXY-BSQD2023022)

Natural Science Foundation of Henan Province(242300420655)

RIGHTS & PERMISSIONS

Shanghai University

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