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A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diﬀusion and diﬀusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the ﬁne grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin ﬁnite element method for the space derivatives and the ﬁnite diﬀerence scheme for the time Caputo derivative with order α ∈ (1 , 2) and α 1 ∈ (0 , 1). Numerical stability and optimal error estimate O ( h r +1 + H 2 r +2 + τ min { 3 − α, 2 − α 1 } ) in L 2 -norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and ﬁne grid mesh size, respectively. Finally, numerical experiments are provided to conﬁrm our theoretical results and eﬀectiveness of the proposed algorithm.