Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

Purpose – This paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank – Nicholson method while the second step is the second-order BDF method. Design/methodology/approach – A rigorous error analysis is done, and optimal L 2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study ’ s theoretical analysis. Findings – Optimal L 2 error estimates and energy norm. Originality/value – Thegoalofthisresearcharticleistopresentandestablishtheunconditionallyoptimalerror estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal errorestimatefortheproposedmethodsisderivedbyusingthetemporal-spatialerrorsplittingtechniques,which splittheerrorbetweentheexactsolutionandthenumericalsolutionintotwoparts,thatis,thetemporalerrorandthespatialerror.Sincethespatialerrorisnotdependentonthetimestep,theboundednessofthenumerical solutioninL ∞ -norm follows an inverse inequality immediately without any restriction on the grid mesh.


Introduction
In this paper, we consider the following parabolic problem with nonlocal nonlinearity: u t À aðlðuÞÞΔu þ αjuj p−2 u ¼ f ðuÞ in Ω 3 ð0; T; uðx; tÞ ¼ 0 on vΩ 3 ð0; T; uðx; 0Þ ¼ u 0 ðxÞ in Ω; where Ω ⊂ R d , d ≥ 1 is again a domain with a smooth boundary vΩ, a and f are functions to be defined in the next section and l denote a continuous linear form on L 2 (Ω) given by where g is a function on L 2 (Ω).

Optimal error estimates of a BDF scheme
The study of nonlocal parabolic problems has received considerable attention in recent years ( [1][2][3] and the references therein).This kind of problems arises in various situations, for instance, u could describe the density of a population (for instance, bacteria) subject to spreading.The diffusion coefficient a is then supposed to depend on the entire population in the domain rather than on the local density, that is, moves are guided by considering the global state of the medium.The problem is nonlocal in the sense that the diffusion coefficient is determined by a global quantity.Besides its mathematical motivation because of the presence of the nonlocal term a(l(u)), such problems come from physical situations related to migration of a population of bacteria in a container in which the velocity of migration v 5 a∇u depends on the global population in a subdomain Ω 0 ⊂ Ω given by a(l(u)).
Simsen and Ferreira [4] have discussed not only the existence and uniqueness of solutions for this problem but also continuity with respect to initial values, the exponential stability of weak solutions and important results on the existence of a global attractor.The numerical methods for the nonlocal problems have been investigated by many authors as like in Refs [5,6] and the references therein.However, they are restricted to nonlocal reaction terms or nonlocal boundary conditions.Chaudhary et al. [7] studied the convergence analysis of the Crank-Nicolson finite element method for the nonlocal problem involving the Dirichlet energy.Mbehou et al. [8] studied (1.1) using the Crank-Nicolson Galerkin finite element method.The main focus on this paper was to present the exponential decay and vanishing of the solutions in finite time.They also derived the optimal convergence order in L 2 -norm using P r with r ≥ 1 finite elements.Yin and Xu [9] applied the finite-volume method to obtain approximate solutions for a nonlocal problem on reactive flows in porous media and derived the optimal convergence order in the L 2 norm.Almeida et al. [10] presented convergence analysis for a fully discretized approximation to a nonlocal problem involving a parabolic equation with moving boundaries, with the finite element method applied for the space variables and the Crank-Nicolson method for the time.Recently, Yang et al. [11] derived the unconditional optimal error estimate of Galerkin FEMs for the time-dependent Klein-Gordon-Schrodinger equations using the error splitting technique.Also in Ref. [12], Yang and Jiang applied the linearized second-order backward differentiation formulae (BDF) Galerkin Finite element methods (FEMs) for the Landau-Lifshitz equations to derive the unconditional optimal error estimates.
Our goal in this research article is to give and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem (1.1).Using P r (r ≥ 1) finite element to approximate the solution of (1.1), the optimal error estimates O(Δt 2 þ h rþ1 ) in L 2 norm are derived using the error splitting technique.
This paper is organized as follows.In Section 2, we recall few known results and present few regularities, which are used in the proof of the optimal error estimates.To prove the optimal error estimates by the error splitting technique, the temporal errors and the spatial errors are shown in Sections 3 and 4, respectively.Finally numerical results are presented in Section 5 to demonstrate our theoretical analysis.

Preliminaries and main results
Let Ω ⊂ R d (d ≥ 1) be a bounded domain with a smooth boundary vΩ 5 Γ.The standard notations (see for instance Refs [13,14]) will be used throughout this work.The Lebesgue space is denoted L p (Ω), 1 ≤ p ≤ ∞, with norms k$k L p but the L 2 (Ω)-norm will be denoted by k $k.For any nonnegative integer m and real number p ≥ 1, the classical Sobolev spaces: ; and the norm with the usual extension when p 5 ∞.When p 5 2, W m,p (Ω) is the Hilbert space H m (Ω) with the scalar product: The norm of H m (Ω) will be denoted by k $k m .It should be mentioned that D α stands for the derivative in the sense of distribution, while α 5 (α 1 , . .., α d ) denotes a multi-index of length We also employ the standard notation of Bochner spaces, such as L q (0, T, X) with norm where X is an Hilbert space and k $k X the norm of X.For all these notions on Sobolev spaces and Bochner spaces, we refer to Refs [13,15].Throughout this paper, the following known inequalities will be frequently used [13].
∀v ∈ H 2 ðΩÞ: Let us now suppose that α is a nonnegative constant and p > 1. Simsen and Ferreira [4] proved the existence and uniqueness of global solution under the following hypotheses.
Given the hypotheses (H1)-(H5), we will also adopt another hypothesis, namely H6. for all r ≥ 1, The following lemmas will be useful.
Lemma 2.4 (H k -estimate of elliptic equations [18]).Suppose that v is a solution of the boundary value problem where Ω ⊂ R d , d 5 2, 3, is a smooth and bounded domain.Then,

14)
Let T h ¼ fKg be a uniform triangular or tetrahedral partition of Ω into triangles or tetrahedrons.Thus, let h ¼ max K∈T h fh K g denote the mesh size, where h K 5 diam(K) 5 max {kx À yk, x, y ∈ K}, and V h be the finite dimensional subspace of H 1 0 ðΩÞ, which consists of continuous piecewise polynomials of degree r ≥ 1 on T h .
Let {t n j t n 5 nΔt; 0 ≤ n ≤ N} be a uniform partition of [0, T] with time step Δt 5 T/N.We write u n 5 u(x, t n ), U n ≈ u(x, t n ) and for any sequence of functions fw n g N n¼0 define The following telescope formula is for n ≥ 2 (2.15) Under the above notations, we propose the following linearized second-order BDF Galerkin finite element scheme associated to (1.1), which is to find where b (2.17) Step 2: Theorem 2.2 Assume that the hypotheses (H1)-(H5) hold.Then the fully discrete system defined in (2.16)-(2.18)has a unique solution U n h which satisfies Proof. 1 For the existence, taking

Optimal error estimates of a BDF scheme
Drop the third term of the left hand side, use the lower bound of a($) and (H2), and ), the same arguments used above give us Taking v h ¼ U n h in (2.18), using the lower bound of a($), (H2) and dropping the third term of the left hand side lead to From the telescope (2.15), we obtain That is The where C is a positive constant independent of Δt and h.
The proof of this theorem will be done in the following sections.

Error estimates for the semi-discrete problem
Let us introduce the corresponding time discrete system associated with (1.1) Step 1: for U 0 5 u 0 , find U 1 by where b U Step 2: for 2 ≤ n ≤ N, find U n by The weak formulations of (3.1)-(3.3)are defined as follows: find and for 2 The existence and uniqueness of the solution to problems (3.4)-(3.6)can be easily proved by using Lax-Milgram theorem.
Let u be the exact solution of (1.1).Then, u satisfies the following equations: , R 1 and R n are, respectively, the truncation errors given by b R By Taylor formula and relation (2.9) with τ 5 1, it is easy to see that Let us denote We have the following assumption.
Lemma 3.1 Assume that the exact solution u of (1.1) satisfies the regularities (2.8).Then there exists a positive constant C independent of Δt such that Proof.Subtracting (3.6) from (3.9) leads to Testing the above equation by b e Using the left bound of a($) to the left hand side and Young's inequality to the right hand side, we obtain The proof ended by dropping the third term of the left hand side and applying (3.10) to the right hand side.
Based upon (3.11), we have AJMS Proposition 3.1 Suppose that the solution u of (1.1) satisfies the regularities (2.8).Then there exists a generic constant C that does not dependent on Δt such that Proof.Subtracting (2.16) from (3.7) and observing that e 0 5 0 leads to Testing the above equation by e 1 and using the fact that e 1 ¼ 1 2 e 1 , we have We have Taking these estimates into (3.13),we obtain the desire result.
The main result in this section is as follows.
Theorem 3.1 Suppose that the solution u of (1.1) satisfies the regularities (2.8).Then there exists a generic constant C that does not dependent on Δt such that Optimal error estimates of a BDF scheme where C is a positive constant independent of n and Δt.
Proof.The proof of this theorem will be done using the mathematical induction.In view of (3.11) and (3.12), the inequality (3.14) holds for n 5 0, 1.Since U 0 5 u 0 , the inquality (3.15) holds for n 5 0. Now, let us assume that (3. Multiply (3.17) by 4Δte n and integrate it over Ω.The use of the telescope formula to the resulting equation leads to Use the lower bound of a($) and drop certain positive terms on the left hand side of the above equation leads to (3.17) We have Summing up the above inequality and using the discrete Gronwall inequality, we get From ke n k ≤ CΔt 2 , we have Applying Lemma 2.4 for the linear elliptic problems (3.1)-(3.3)with the induction assumptions gives the H 2 estimate 3), we have

Error estimates for the fully discrete problem
In this section, we will prove the optimal spatial error estimates.Let Π h be an interpolation operator and R h : H 1 0 Ω ð Þ→ V h be a Ritz projection operator defined by Then we have the following lemma.
where C is a positive constant that does not depend on h and r.

Let us denote
From lemma 4.1, we have Optimal error estimates of a BDF scheme Lemma 4.2 Assume that the exact solution u of (1.1) satisfies the regularities (2.8).Then there exists a positive constant C independent of Δt and h such that Proof.From equations (2.17) and (3.6), b e 1 h satisfies the following equation: From (4.5) and (4.6), we have Combining these estimates into (4.9),we get (4.7).

AJMS
From the inverse inequality, kb e The main result in this section is as follows.
Theorem 4.1 Suppose that the exact solution u of (1.1) satisfies the regularities (2.8).Then there exists a positive constant C independent of Δt and h such that Proof.The proof of this result will be done by mathematical induction.Since U 0 h ¼ Π h U 0 , (4.11) holds for n 5 0. To compute the error estimate (4.10) for n 5 1, subtract (3.4) from (2.16) and take From (4.5) and (4.6), we have Taking these estimates into (4.12) and using Lemma 4.2, we conclude that Optimal error estimates of a BDF scheme If one takes v h ¼ 4Δte n h and uses the telescope formula, one obtains The quantities K i , i 5 1, . .., 5 can be bounded by the similar way T i , i 5 1, . .., 5: Taking these bounds into (4.14),we obtain Sum up the above inequality and use the discrete Gronwall Lemma 2.3 leads to

Numerical results
In this section, we present several numerical simulations to illustrate our theoretical analysis.
Since the resulting matrix of the linear system (2.16)-(2.18) is sparse, symmetric and positive definite, an incomplete Cholesky factorization is performed and the result is used as preconditioner in the preconditioned conjugate method iterative solver (see for instance Refs [20,21]).
To analyze the convergence rate, we consider the following problem.g is chosen correspondingly to the exact solution uðx; y; tÞ ¼ 2ð1 þ t 2 expð−tÞÞxyð1 À xÞð1 À yÞ: We simulated the above problem on uniform meshes with a linear finite element approximation (r 5 1) and T 5 0.1.
For the convergence with respect to the mesh size h, we choose Δt 5 h 2 and we solve problem (2.16)-(2.18)with different values of h (h 5 1/5; 1/10; 1/15; 1/20; 1/25); from our theoretical analysis, the L 2 -norm errors are in order O(h 2 þ Δt 2 ) 5 O(h 2 þ h 4 ) ∼ O(h 2 ).H 1 -norm errors are in order O(h þ Δt 2 ) 5 O(h þ h 4 ) ∼ O(h).In Figure 1, we plot the log of errors against log(h).One can see that for L 2 -norm, the slope is almost 2, and for H 1 À norm, the slope is almost 1, which are in good agreement with our theoretical analysis.

Conclusion
We have presented and analyzed a linearized second-order BDF Galerkin finite element method for the nonlocal parabolic problems.We have proved the L 2 and energy error estimates using sufficient conditions on the exact solution.We also presented some numerical  experiments on Matlab's environment, and our numerical results confirm the theoretical analysis.The results in this paper lay the foundation for developing finite element based numerical methods for more general and complicated nonlocal problems both stationary and evolutionary.

1 h
16)-(2.18), respectively, the existence and uniqueness of U 1 h , b U and U n h are from the Lax-Milgram theorem and the hypothesis (H3).
14) and (3.15) hold for n ≤ m with m ≤ N À 1. Then we need to prove the inequality for n 5 m þ 1.By the definition of b U n and the induction assumption, k b U n k ∞ ≤ C. Subtracting (2.18) from (3.8) results in the following equation: 13) which proves (4.10) for n 5 1.Now, we assume that (4.10) and (4.11) hold for n 5 m À 1, 2 ≤ m ≤ N, then we need to show it also holds for n 5 m.By the definition of b U n h and the induction assumption, k b U n h k ∞ ≤ C. Subtracting (2.18) from (3.5), we obtain

Figure 1 .
Figure 1.Convergence rate with respect to the mesh size h in L 2 and H 1 norm

Figure 2 .
Figure 2. Convergence rate with respect to the time step Δt in L 2 norm relation(2.19) is obtained by summing up the above relation (2.21) and using the discrete Gronwall lemma 2.3.The main result of this work is presented in the following theorem.