Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations

We consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition $v(x)$ and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain Ω. We study two semidiscrete approximation schemes, i.e., the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions. We establish almost optimal with respect to the data regularity error estimates, including the cases of smooth and nonsmooth initial data, i.e., v ∈ H²(Ω)∩ H¹₀(Ω) and v ∈ L₂(Ω). For the lumped mass method, the optimal L₂-norm error estimate is valid only under an additional assumption on the mesh, which in two dimensions is known to be satisfied for symmetric meshes. Finally, we present some numerical results that give insight into the reliability of the theoretical study.