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<p>This paper presents a novel and efficient approach of nonuniform L1-based two-grid algorithm for solving nonlinear multi-term time-fractional diffusion (MTTFD) equation with variable coefficients. To tackle the singularity issue of solution at initial time, a temporal L1 scheme is adopted on graded meshes. The computational expense is decreased by introducing a new two-grid technique based on finite element method (FEM). Using fine-grid solutions from each historical time level to address the nonlinear MTTFD equation on a coarser grid, the algorithm removes the requirement for redundant discrete kernel summations across coarse and fine grids. Furthermore, an innovative approach for calculating the kernels is introduced in a simple and efficient manner to prevent roundoff errors. The inclusion of a fast nonuniform L1 formula in two-grid FEM accelerates Caputo derivative evaluations. The stability analysis, along with optimal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-error estimates is rigorously derived for both the L1-FEM and two-grid L1-FEM approaches. These error estimates are shown to be robust even when the highest fractional derivative order is varied. The existence and uniqueness of a fully discrete solution to two-grid L1-FEM algorithm are also rigorously established. Finally, the proposed two-grid FEM algorithm is validated through numerical experiments that support the theoretical results, showcasing the robustness, superior efficiency, and accuracy of the algorithm.</p>

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A nonuniform and fast two-grid L1-FEM algorithm for nonlinear multi-term time-fractional diffusion equation and its robust error analysis

  • 期刊论文

  • 发表日期：

    2025年12月24日

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  • Zhijun Tan

摘要

This paper presents a novel and efficient approach of nonuniform L1-based two-grid algorithm for solving nonlinear multi-term time-fractional diffusion (MTTFD) equation with variable coefficients. To tackle the singularity issue of solution at initial time, a temporal L1 scheme is adopted on graded meshes. The computational expense is decreased by introducing a new two-grid technique based on finite element method (FEM). Using fine-grid solutions from each historical time level to address the nonlinear MTTFD equation on a coarser grid, the algorithm removes the requirement for redundant discrete kernel summations across coarse and fine grids. Furthermore, an innovative approach for calculating the kernels is introduced in a simple and efficient manner to prevent roundoff errors. The inclusion of a fast nonuniform L1 formula in two-grid FEM accelerates Caputo derivative evaluations. The stability analysis, along with optimal \(L^{2}\) L 2 - and \(H^{1}\) H 1 -error estimates is rigorously derived for both the L1-FEM and two-grid L1-FEM approaches. These error estimates are shown to be robust even when the highest fractional derivative order is varied. The existence and uniqueness of a fully discrete solution to two-grid L1-FEM algorithm are also rigorously established. Finally, the proposed two-grid FEM algorithm is validated through numerical experiments that support the theoretical results, showcasing the robustness, superior efficiency, and accuracy of the algorithm.