Numerical Solution of Fractional Diffusion and Wave Equations

In the past few years, the numerical approximation of fractional diffusion and wave equations has attracted the attention of many researchers, and recently, a lot of progress has been made in this direction. In this project, we propose and analyzed a finite volume element method (FVEM) for solving fractional diffusion and wave equations in a two-dimensional convex domain. A numerical solution is found by applying both piecewise-constant and piecewise-linear discontinuous Galerkin (DG) methods in time. Both semi-discrete and completely discrete will be analyzed. The effect of numerical quadrature will be examined as well. Since the solution has a singular behavior near t = 0, the time mesh is graded appropriately near t = 0. Optimal error estimates will be obtained and some super-convergence results will be discussed. Special attention will be given to the regularity of exact solution.