Unconditionally optimal convergence of a linearized Galerkin FEM for the nonlinear time-fractional mobile/immobile transport equation

Abstract In this paper, a linearized Galerkin finite element method (FEM) is discussed for solving the nonlinear time-fractional mobile/immobile transport equation. Utilizing the temporal–spatial error splitting argument, we derive the optimal L 2 -norm error estimate without the stepsize restriction condition τ = O ( h d 4 ) . The key point in our analysis is to obtain the unconditionally optimal error estimate between the solutions of the time-discrete system and continuous problem in H 2 -norm, with which, we prove the boundedness of the fully discrete finite element solution in L ∞ -norm by using induction method. Then, the unconditionally optimal error estimate in L 2 -norm can be obtained in the usual way. Finally, three numerical examples in both two and three dimensional spaces are given to illustrate the correctness of our theoretical analysis.