turkmath.org

We consider an initial boundary value problem for the inhomogeneous time-dependent fractional diffusion equations with homogeneous Dirichlet boundary condition in a bounded convex polyhedral domain. We introduce and study the semi-discrete approximations based on the standard Galerkin and lumped mass finite element methods. We derive almost optimal estimate when the right hand side of the equation $f (x,t)$ is in the space $L_\infty((0,T), \dot{H}^s), -1 < s\leq 1$. For lumped mass method, the optimal $L_2$-norm error estimate is established for meshes that have a symmetry property. Finally, we present numerical experiments that confirm the theoretical results. (This is based on my joint research with Bangti Jin, Joseph Pasciak, and Zhi Zhou, all from Texas A&M University)