2014/09/05 by Kassem Mustapha, Mustapha, Kassem, B. Abdallah +3 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1409.1935
openalex publication_date 2014/09/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We propose and analyze a time-stepping discontinuous Petrov-Galerkin method\ncombined with the continuous conforming finite element method in space for the\nnumerical solution of time-fractional subdiffusion problems. We prove the\nexistence, uniqueness and stability of approximate solutions, and derive error\nestimates. To achieve high order convergence rates from the time\ndiscretizations, the time mesh is graded appropriately near~t=0 to compensate\nthe singular (temporal) behaviour of the exact solution near t=0 caused by\nthe weakly singular kernel, but the spatial mesh is quasiuniform. In the\nL_\∞((0,T);L2(\Ω))-norm ((0,T) is the time domain and \Ω is\nthe spatial domain), for sufficiently graded time meshes, a global convergence\nof order km+\α/2+hr+1 is shown, where 0<\α<1 is the\nfractional exponent, k is the maximum time step, h is the maximum diameter\nof the spatial finite elements, and m and r are the degrees of approximate\nsolutions in time and spatial variables, respectively. Numerical experiments\nindicate that our theoretical error bound is pessimistic. We observe that the\nerror is of order ~km+1+hr+1, that is, optimal in both variables.\n