<p>In this paper, a second-order backward differentiation formula (BDF2) <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-Galerkin mixed finite element method (FEM) is developed for solving the nonlinear Kirchhoff-type equation with a damping term. By introducing a new variable <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\varvec{{q}}}=\nabla u_t+(1+\Vert \nabla u\Vert ^2)\nabla u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>u</mi> <mi>t</mi> </msub> <msup> <mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O(h^2+\tau ^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <i>u</i> in the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\nabla \cdot {\varvec{{q}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm are derived in detail. At last, a numerical example is carried out to illustrate the correctness of the theoretical analysis.</p>

Superconvergence Analysis of an $H^1$-Galerkin Mixed FEM for Nonlinear Kirchhoff-Type Equation with Damping Term

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发表日期：

2026年6月30日

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Lijuan Guo,
Zhen Guan,
Xinyu Wei,
Keke Zhang

摘要

In this paper, a second-order backward differentiation formula (BDF2) $H^1$ $H^{1}$ -Galerkin mixed finite element method (FEM) is developed for solving the nonlinear Kirchhoff-type equation with a damping term. By introducing a new variable ${\varvec{{q}}}=\nabla u_t+(1+\Vert \nabla u\Vert ^2)\nabla u$ $q = \nabla u_{t} {+ (1 + ‖ \nabla u ‖}^{2}) \nabla u$ , 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 $O(h^2+\tau ^2)$ $O (h^{2} + τ^{2})$ of u in the $H^1$ $H^{1}$ -norm and $\nabla \cdot {\varvec{{q}}}$ $\nabla \cdot q$ in the $L^2$ $L^{2}$ -norm are derived in detail. At last, a numerical example is carried out to illustrate the correctness of the theoretical analysis.

Superconvergence Analysis of an \(H^1\)-Galerkin Mixed FEM for Nonlinear Kirchhoff-Type Equation with Damping Term

摘要