Analysis of stability and convergence for L-type formulas combined with a spatial finite element method for solving subdiffusion problems

Abstract

Abstract. A time-fractional diffusion equation with the Caputo fractional derivative of order α ∈ (0, 1) is considered on a bounded polygonal domain. Some numerical methods are presented based on the finite element method (FEM) in space on a quasi-uniform mesh and L-type discretizations (i.e., L1, L1-2, and L1-2-3 formulas) to approximate the Caputo derivative. Stability and convergence of the L1-2-3 FEM as well as L1-2 FEM are proved rigorously. The lack of positivity of the coefficients of these formulas is the main difficulty in the analysis of the proposed methods. This has hampered the analysis of methods using finite elements mixed with L1-2 and L1-2-3 discretizations. Our proofs are based on the concept of a special kind of discrete Grönwall’s inequality and the energy method. Numerical examples confirm the theoretical analysis.