A stabilizer-free weak Galerkin finite element method for an optimal control problem of a time fractional diffusion equation

Abstract

This paper proposes a fully discrete stabilizer-free weak Galerkin (SFWG) finite element approximation for an optimal control problem driven by a time fractional diffusion equation. We focus on the spatial discretization of the SFWG finite element approximation to establish a semi-discrete scheme, followed by the application of L1 discretization to the Caputo fractional derivative in time to derive a fully discrete scheme. We then prove a priori error estimate for the discrete schemes. To reduce the computational complexity, we incorporate the proper orthogonal decomposition (POD) technique on the state and adjoint state systems to obtain a fully discrete reduced-order stabilizer-free weak Galerkin (ROSFWG) finite element method. Finally, the validity of the theoretical analysis is confirmed through numerical experiments.