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<p>In this paper, a distributed-order time-fractional equation with variable coefficients on polygonal meshes is considered. Initially, the distributed-order time-fractional derivative is converted into a multi-term time-fractional derivative. The L1 scheme on graded meshes is then employed to handle the singularity of the time-fractional derivative. For variable coefficients in the spatial direction, modified approximated bilinear forms of the virtual element method are constructed to maintain optimal convergence. Utilizing the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-robust discrete fractional Grönwall inequality, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-robust stability and optimal error estimates of the fully discrete scheme in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm are established. These results remain valid and stable as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\beta \rightarrow 1^-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Numerical results are presented to illustrate the sharpness of the theoretical findings.</p>

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\(\beta \)-robust virtual element method for distributed order time-fractional reaction-diffusion equation with variable coefficients

  • 期刊论文

  • 发表日期：

    2025年4月9日

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  • Yanping Chen,
  • Jian Lei,
  • Qiling Gu,
  • Jianwei Zhou,
  • Fangfang Qin

摘要

In this paper, a distributed-order time-fractional equation with variable coefficients on polygonal meshes is considered. Initially, the distributed-order time-fractional derivative is converted into a multi-term time-fractional derivative. The L1 scheme on graded meshes is then employed to handle the singularity of the time-fractional derivative. For variable coefficients in the spatial direction, modified approximated bilinear forms of the virtual element method are constructed to maintain optimal convergence. Utilizing the \(\beta \) β -robust discrete fractional Grönwall inequality, \(\beta \) β -robust stability and optimal error estimates of the fully discrete scheme in the \(L^2\) L 2 -norm are established. These results remain valid and stable as \(\beta \rightarrow 1^-\) β → 1 - . Numerical results are presented to illustrate the sharpness of the theoretical findings.