H1-norm Analysis of an Integral-Averaged L1 Method on Nonuniform Time Meshes for a Time-Fractional Diffusion Problem

A time-fractional diffusion problem with a Caputo time-fractional derivative of order α∈(0,1) is considered, the solution of which is typically weakly singular at the initial time. For this problem, we give an H1-norm analysis of the stability and convergence of an integral-averaged L1 method on nonuniform time meshes. The averaging of the L1 scheme that we use is known as the L1+ or L1¯ scheme. A new positive definiteness result for the integral-averaged L1 fractional-derivative operator is established. It improves the previous positive definiteness results in the literature and plays an important role in the analysis of H1-norm error of the integral-averaged L1 method. The H1-norm stability holds for the general nonuniform time meshes, while the H1-norm convergence is proved for the time graded meshes and the H1-norm convergence order in time is min{3+α,γα}/2 for all α∈(0,1), where γ≥1 is the mesh grading parameter. Two full discretization methods using finite differences and finite elements in space are considered. The theoretical results are illustrated by numerical results.