A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

Toprakseven Ş., Dinibutun S.

AIMS Mathematics, cilt.8, sa.12, ss.31022-31049, 2023 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 8 Sayı: 12
  • Basım Tarihi: 2023
  • Doi Numarası: 10.3934/math.20231588
  • Dergi Adı: AIMS Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Directory of Open Access Journals
  • Sayfa Sayıları: ss.31022-31049
  • Anahtar Kelimeler: graded temporal meshes, optimal rate of convergence, stabilizer-free weak Galerkin finite element method, sub-diffusion problems
  • Yozgat Bozok Üniversitesi Adresli: Evet

Özet

We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov’s higher order L2-1σ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in H1 norm and L2 norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant r, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order O(hk+1 + M−2) in the L∞ (L2)-norm has been established. We give several numerical examples to confirm the theory presented in this work.