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<p>This study proposes a high-precision finite element method (FEM) for nonlinear two-dimensional space-time-fractional diffusion equations, combining the Galerkin spatial discretization with the L2-<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1_\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>1</mn> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> temporal scheme. By rigorously analyzing unconditional stability and deriving error estimates, the method achieves spatial convergence of order 2 and temporal convergence of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(3-\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>-</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, validated through numerical experiments under diverse initial conditions. Compared to existing works focusing on linear systems or single-term fractional dynamics, this research innovatively extends the L2-<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(1_\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>1</mn> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> scheme to complex nonlinear space-time-coupled problems, demonstrating its capability to simultaneously handle nonlinearities and fractional derivatives while maintaining computational efficiency and geometric adaptability.</p>

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Finite Element Analysis of Nonlinear 2D Space-Time-Fractional Diffusion Equations Using the \(\hbox {L2-1}_{\sigma }\) Discretization Scheme

  • 期刊论文

  • 发表日期：

    2026年5月29日

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  • Huiqin Zhang,
  • Yanping Chen,
  • Yang Wang,
  • Jian Huang

摘要

This study proposes a high-precision finite element method (FEM) for nonlinear two-dimensional space-time-fractional diffusion equations, combining the Galerkin spatial discretization with the L2- \(1_\sigma \) 1 σ temporal scheme. By rigorously analyzing unconditional stability and deriving error estimates, the method achieves spatial convergence of order 2 and temporal convergence of order \(3-\alpha \) 3 - α , where \(\alpha \in (0,1)\) α ∈ ( 0 , 1 ) , validated through numerical experiments under diverse initial conditions. Compared to existing works focusing on linear systems or single-term fractional dynamics, this research innovatively extends the L2- \(1_\sigma \) 1 σ scheme to complex nonlinear space-time-coupled problems, demonstrating its capability to simultaneously handle nonlinearities and fractional derivatives while maintaining computational efficiency and geometric adaptability.