1.
Introduction
In this paper, we consider the following semilinear parabolic integro-differential equations:
where Ω⊂R2 is a rectangle with boundary ∂Ω, X=(x,y), J=(0,T], f(⋅) is twice continuously differentiable and u0 is a given function. We assume that
There exists a lot of numerical methods for solving nonlinear partial differential equations in the literature. For example, Cannon and Lin [1] derived a priori error estimates of semidiscrete and Crank-Nicolson finite element approximations to the solution of the nonlinear diffusion equations with memory. Eriksson and Johnson [3] used adaptive finite element method to solve nonlinear parabolic problems. Moore [9] considered a posteriori error estimates for semi- and fully discrete finite element methods using a degree polynomial basis for solving nonlinear parabolic equations. Garcia [4] discussed a priori error estimates of fully discrete Raviart-Thomas mixed finite element scheme for nonlinear parabolic equations.
Two-grid method was first proposed by Xu [11,12] as an efficient discretization technique for solving the nonlinear and the nonsymmetric problems. Liu et al. exploited the two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations in [8]. Chen et al. [2] presented a two-grid scheme of mixed finite element method for fully nonlinear reaction-diffusion equations. Hou et al. [5] presented a two-grid method of P20-P1 mixed finite element method combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations. Shi and Mu [10] discussed some superclose results of a two-grid finite element method for semilinear parabolic equations. Yang and Xing [13] discussed the convergence of two-grid discontinuous Galerkin scheme for a kind of nonlinear parabolic problems.
This paper is motivated by the ideas of the works [10,11], we present a two-grid scheme for semilinear parabolic integro-differential equations discretized by finite element method combined with Crank-Nicolson scheme. We mainly discuss the superclose estimates between the numerical solution and the interpolation.
The plan of this paper is as follows. In Section 2, we give the Crank-Nicolson scheme and deduce the superclose result of order O(h2+(△t)2) in the H1-norm. In Section 3, we present the two-grid method and derive the superclose estimates of order O(H2+(△t)2) and order O(h2+H4+(△t)2), respectively. In Section 4, we present a numerical example to demonstrate the effectiveness of our method.
2.
Superclose analysis of the Crank-Nicolson scheme
We adopt the standard notation Wm,p(Ω) for Sobolev spaces on Ω with a norm ‖⋅‖m,p given by ‖v‖pm,p=∑|α|≤m‖Dαv‖pLp(Ω), a semi-norm |⋅|m,p given by |v|pm,p=∑|α|=m‖Dαv‖pLp(Ω). We set Wm,p0(Ω)={v∈Wm,p(Ω):v|∂Ω=0}. For p=2, we denote Hm(Ω)=Wm,2(Ω), Hm0(Ω)=Wm,20(Ω), and ‖⋅‖m=‖⋅‖m,2, ‖⋅‖=‖⋅‖0,2.
We denote by Ls(J;Wm,p(Ω)) the Banach space of all Ls integrable functions from J into Wm,p(Ω) with norm ‖v‖Ls(J;Wm,p(Ω))=(∫T0||v||sWm,p(Ω)dt)1s for s∈[1,∞), and the standard modification for s=∞. For simplicity of presentation, we denote ‖v‖Ls(J;Wm,p(Ω)) by ‖v‖Ls(Wm,p). Similarly, one can define the spaces H1(J;Wm,p(Ω)). In addition C denotes a general positive constant independent of h and △t, where h is the spatial mesh-size, and △t is the time step.
Let Th be a uniform rectangular partition of Ω with mesh size h. Vh be the bilinear finite element space with vanishes on ∂Ω. Let Ih and Rh be the associated interpolation and Ritz projection operators on Vh, respectively(see[10]).
Then, for u∈H10(Ω)∩H3(Ω), from [10] we know that
The weak formulation of (1) is to find u:J→H10(Ω), such that
Let {tn⏐tn=n△t;0≤n≤N} be a uniform partition in time with time step △t, un=u(X,tn) and tn−1/2=(tn−1+tn)/2. For a sequence of functions {ϕn}Nn=0, we denote dtϕn=(ϕn−ϕn−1)/△t, then the Crank-Nicolson scheme of (1) is to find unh∈Vh for n=1,2,...,N, such that
For the proof of existence and uniqueness of the solution for the nonlinear algebraic problem (9)-(10), please refer to [6].
Theorem 2.1. Let u and unh be the solutions of (8) and (9), respectively. Assume that u∈L∞(H3), ut∈L2(H2), utt∈L2(L2), uttt∈L2(L2), ∇utt∈L2(H2)∩L∞(L2) and ∇uttt∈L2(L2), then, for n=1,2,…,N, we have
Proof. Letting t=tn−1/2 and v=vh in (8), we get
where Rn1=dtun−un−1/2t, Rn2=un+un−12−un−1/2.
Setting un−unh=un−Rhun+Rhun−unh:=ηn+ξn. Subtracting (9) from (12), with the help of (7), we have
Selecting vh=dtξn in (13), noting that
then multiplying (13) by 2△t and summing from n=1,...,l(1≤l≤N), we conclude that
where we used (7) and ξ0=0.
Now, we estimate the right-hand terms of (15). For I1, it is easy to check that
which together with Cauchy inequality, Young's inequality and (4) yields
For I2, we decompose it as
Using Cauchy inequality and Young's inequality, we see that
and
Thus, we get
For I3, by virtue of Cauchy inequality and Young's inequality, we have
where we used the following estimate
where ϕn is located between tn and tn−1/2, ψn is located between tn−1/2 and tn−1, λn is located between ϕn and ψn, and
Next, we estimate I4. By use of mean value theorem and the assumption on f, we conclude that
where we also used
and
where θn1 is located between tn−1 and tn−1/2, θn2 is located between tn−1/2 and tn, θn3 is located between θn1 and θn2, λn1 is located between un−1 and un−1/2, λn2 is located between un−1/2 and un, λn3 is located between λn1 and λn2.
Using (4), mean value theorem and the assumption on f, we easily get
By use of Cauchy inequality, Young's inequality and (24)-(25), we derive
For I5, from Cauchy inequality, Young's inequality and
we have
where ρn is located between tn−1 and tn.
Notice that
Using Taylor expansion, we know that
where tn<βn<tn+1/2<γn<tn+1.
Using Cauchy inequality, Young's inequality, (29), (30) and mean value theorem, we get
where λl is located between tl−1 and tl, δn is located between tn−1/2 and tn+1/2.
For I6, combining Cauchy inequality, Young's inequality, (28) with (31)-(32), we derive
Now, substituting the estimates for I1-I6 into (15), then applying discrete Gronwall's lemma, for sufficiently small △t, we have
which together with (5), Poincare's inequality and triangle inequality yields (11). We complete the proof of the theorem.
3.
Superclose analysis of the two-grid scheme
In this section, we present the main algorithm of the paper, which has the following two steps:
Step 1. On the coarse grid TH, compute u1H∈VH to satisfy the following original nonlinear system:
For n=1,...,N−1, compute un+1H∈VH to satisfy the following linear system:
Step 2. On the fine grid Th, for n=1,...,N, compute ˜unh∈Vh to satisfy the following linear system:
Now, we shall discuss the superclose estimates of the above two-grid algorithm in the following theorem.
Theorem 3.1. Let u, unH and ˜unh be the solutions of (8), (35)-(37) and (35)-(39), respectively. Then under the conditions of Theorem 2.1, for n=1,2,…,N, we have
Proof. Setting un−unH=un−RHun+RHun−unH:=ωn+φn. From Theorem 2.1, (40) is obvious for n=1.
For n=1,...,N−1, letting t=tn and v=vH in (8), we get
where Rn3=un+1−un−12△t−unt, Rn4=un+1+un−12−un.
Subtracting (37) from (42), with the help of (7), we have
Selecting vH=φn+1−φn−1△t in (43), using φ0=0 and the equality
then multiplying the resulting equation by 2△t and summing from n=1,...,l(1≤l≤N−1), we conclude that
where we used (7) and φ0=0.
Similar to the estimates of I1-I6, we can estimate D1-D6 as
At last, for D7, using (5), (40) and triangle inequality, we see that
Now, substituting the estimates for D1-D7 into (45), then applying discrete Gronwall's lemma, for sufficiently small △t, we have
which together with (5), Poincare's inequality and triangle inequality yields
Using Taylor expansion, we have
where αn is located between un and unH.
Setting Rhun−˜unh:=˜ξn. Subtracting (38) from (12), similar to (15), we conclude that
Now, we estimate B1-B7, respectively. For B3, similar to (24), we know that
For B4, we find from Cauchy inequality and the assumption on f that
Using Young's inequality and (4), we know that
Combining (40), Cauchy inequality, Young's inequality and interpolation theory with H1↪L4, we derive
Now, from (59)-(60), we find that
Similarly, we can estimate B5 as
Similar to (16), (21), (27) and (33), we easily have
It follows from (56)-(57), (61)-(63) and Poincare's inequality that
Thus, for sufficiently small △t, using discrete Gronwall's lemma and Poincare's inequality, we arrive at
which together with (5) and triangle inequality yields
The proof is complete.
4.
Numerical experiments
In this section, we are going to validate the superclose estimates for two-grid discretization method for semilinear parabolic integro-differential equations by a concrete numerical example.
We consider the following semi-linear parabolic integro-differential equations
where Ω=(0,1)2 and J=(0,1]. We choose u(X,t)=sin(πt)sin(πx1)sin(πx2) as the exact solution. Then, the explicit formulation of g(X,t) is
We first test the example for the Crank-Nicolson scheme. The error and the convergence order of ‖unh−Ihun‖1 at t=0.125 with h=Δt are presented in Table 1. Obviously, it is the same with the result in Theorem 2.1. Next, the two-grid scheme is tested. The error and the convergence order of ‖unH−IHun‖1 and ‖˜unh−Ihun‖1 are provided in Table 2 and Table 3. We find from these two tables that the result coincides with that in Theorem 3.1. Finally, we show the efficiency of the two-grid method by comparing the cpu time in Table 4.