Piecewise Linear Discontinuous Petrov Galerkin Method for Time Fractional Diffusion Equations

Abstract

We propose and analyze piecwise linear discontinuous Petrov-Galerkin method in time combined with a standard conforming finite element method in space for the numerical solution of time-fractional diffusion problems of order 0 < μ < 1. We prove the stability of the exact solution. The existence, uniqueness and stability of approximate solutions will be proved. We employ a non-uniform mesh based on concentrating the cells near the singularity. The advantage of employing a non-uniform mesh is improving the accuracy of the approximate solution. Numerical experiments indicate the error in L∞(0, T ; L2(Ω))-norm is of order kmin(γ(1-  μ),2) + h2, where k denotes the maximum time steps and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh and γ > 0.