Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

and Applied Analysis 3 Here B(I × Ω) is a Banach space with respect to the following norm: ‖V‖Bα/2(I×Ω) = (‖V‖ 2 Hα/2(I,L2(Ω)) + ‖V‖ 2 L2(I,H 1 0 (Ω)) ) 1/2 , (15) whereH(I, L2(Ω)) :={V; ‖V(t, L2(Ω) ∈ H α/2 (I)}, endowed with the norm ‖V‖Hα/2(I,L2(Ω)) := 󵄩󵄩󵄩󵄩 ‖V (t, L2(Ω) 󵄩󵄩󵄩󵄩Hα/2(I) . (16) Based on the relation equation between the left Caputo and the Riemann-Liouville derivative in [31], we can translate the Caputo problem to the Riemann-Liouville problem. Then, we consider the weak formulation of (1) as follows. For f ∈ B α/2 (I × Ω) , find u(t, x) ∈ B(I × Ω) such that A (u, V) = F (V) , V ∈ B α/2 (I × Ω) , (17) where the bilinear form is, by Lemma 1, A (u, V) := ( R 0D α/2 t u, R t D α/2 T V) L2(I×Ω) + s ∑ i=1 ai( R 0D αi/2 t u, R t D αi/2 T V) L2(I×Ω) + (∇xu, xL2(I×Ω) , (18) and the functional isF(V) := (f, V)L2(I×Ω),f(t, x) := f(t, x)+ (u0(x)t −α /Γ(1 − α)) + ∑ s i=1 ai(u0(x)t i/Γ(1 − αi)). Based on themain results in Subsection 3.2 in [32], we can prove the following existence and uniqueness theorem. Theorem 2. Assume that 0 < α < 1 and f ∈ B(I × Ω). Then the system (17) has a unique solution in B(I × Ω). Furthermore, ‖u‖Bα/2(I×Ω) ≲ 󵄩󵄩󵄩󵄩 f 󵄩󵄩󵄩󵄩Bα/2(I×Ω)󸀠 . (19) Proof. The existence and uniqueness of the solution of (17) is guaranteed by the well-known Lax-Milgram theorem. The continuity of the bilinear form A and the functional F is obvious. Now we need to prove the coercivity of A in the space B(I × Ω). From the equivalence of H 0 (I × Ω), r H α 0 (I × Ω) andH 0 (I × Ω), for all u, V ∈ B(I × Ω), using the similar proof process in [32], we obtain A (V, V) ≳ ( R 0D α/2 t V, R 0 D α/2 t V) L2(I×Ω) + s ∑ i=1 ai( R 0D αi/2 t V, R 0 D αi/2 t V) L2(I×Ω) + (∇xu, xL2(I×Ω) ≳ ‖V‖ 2 B(I×Ω) . (20) Then we take V = u in (17) to get ‖u‖ B(I×Ω) ≲ (f, L2(I×Ω) by the Schwarz inequality and the Poincaré inequality. 3. Time Discretization and Convergence In this section, we consider DFBDM for the time discretization of (1)–(3), which is introduced in [33] for fractional ordinary differential equations. We can obtain the convergence order for the time discretization for theMT-FPDEs. Let A = −Δ x, D(A) = H 1 0 (Ω) ∩ H 2 (Ω). Let u(t), f(t), and u(0) denote the one-variable functions as u(t, ⋅), f(t, ⋅), and u(0, ⋅), respectively. Then (1) can be written in the abstract form, for 0 < t < T, 0 < αs < ⋅ ⋅ ⋅ < α1 < α < 1, with initial value u(0) = u0. Now we have R 0 D α t [u − u0] (t) + s ∑ i=1 ai R 0 D αi t [u − u0] (t) + Au (t) = f (t) . (21) Let 0 = t0 < t1 < ⋅ ⋅ ⋅ < tN = T be a partition of [0, T]. Then, for fixed tj, j = 1, 2, . . . , N, we have R 0 D α t [u − u0] (tj) = t −α j