Novel superconvergence analysis of a low order FEM for nonlinear time-fractional Joule heating problem

Abstract

The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L1 method. First, a novel superclose estimate in the H1-norm is derived rigorously with some new analysis techniques under low regularity of the solutions un,ϕn∈L∞(0,T;H3(Ω)) rather than un∈L∞(0,T;H4(Ω)) and ϕn∈L∞(0,T;H3(Ω)∩W2,∞(Ω)) required in the previous studies. Then, the global superconvergence result is deduced by interpolated post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis. It should be mentioned that the analysis and results presented herein are also valid to some other known conforming and nonconforming finite elements.