Arbitrary high-order unconditionally stable methods for reaction-diffusion equations via Deferred Correction: Case of the implicit midpoint rule
aa r X i v : . [ m a t h . NA ] A ug Mathematical Modelling and Numerical Analysis
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ARBITRARY HIGH-ORDER UNCONDITIONALLY STABLE METHODS FORREACTION-DIFFUSION EQUATIONS VIA DEFERRED CORRECTION:CASE OF THE IMPLICIT MIDPOINT RULE
Saint-Cyr E.R. Koyaguerebo-Im´e and Yves Bourgault Abstract . In this paper we analyse full discretizations of an initial boundary value problem (IBVP)related to reaction-diffusion equations. The IBVP is first discretized in time via the deferred correctionmethod for the implicit midpoint rule and leads to a time-stepping scheme of order 2 p + 2 of accuracyat the stage p = 0 , , , · · · of the correction. Each semi-discretized scheme results in a nonlinear ellipticequation for which the existence of a solution is proven using the Schaefer fixed point theorem. Theelliptic equation corresponding to the stage p of the correction is discretized by the Galerkin finite ele-ment method and gives a full discretization of the IBVP. This fully discretized scheme is unconditionllystable with order 2 p + 2 of accuracy in time. The order of accuracy in space is equal to the degreeof the finite element used when the family of meshes considered is shape-regular while an incrementof one order is proven for shape-regular and quasi-uniform family of meshes. A numerical test witha bistable reaction-diffusion equation having a strong stiffness ratio is performed and shows that theorders 2,4,6,8 and 10 of accuracy in time are achieved with a very strong stability. . Introduction
Let Ω be a bounded domain in R d ( d = 1 , ,
3) with smooth boundary ∂ Ω and
T >
0. Consider the followingreaction-diffusion system with Cauchy-Dirichlet conditions u ′ − M ∆ u + f ( u ) = S in Ω × (0 , T ) u = 0 on ∂ Ω × (0 , T ) u ( .,
0) = u in Ω , (1)where u : Ω × [0 , T ] → R J is the unknown, for a positive integer J , M is an J × J constant matrix, f : R J → R J and S : Ω × (0 , T ) → R J are given smooth functions. This is a general form of reaction-diffusion equations (seefor instance [1]) that model various phenomena in physics, combustion, chemical reactions, population dynamicsand biomedical science (cancer modelling and other physiological processes) (see, e.g., [1–5]). Keywords and phrases: time-stepping methods, deferred correction ,high order methods, reaction-diffusion equations, finiteelements Department of Mathematics and Statistics, University of Ottawa, STEM Complex, 150 Louis-Pasteur Pvt, Ottawa, ON,Canada, K1N 6N5, Tel.: +613-562-5800x2103 e-mail: s [email protected], [email protected] c (cid:13) EDP Sciences, SMAI 1999
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We suppose that M is positive definite and the function f satisfies the following two monotonicity conditions( f ( x ) − f ( y ) , x − y ) ≥ α | x − y | q + τ ( y ) | x − y | , ∀ x, y ∈ R J , for some α ≥ , q ≥ , (2)and ( df ( x ) y ) · y ≥ − µ | y | , ∀ x, y ∈ R J , (3)where µ is a nonnegative real, and τ is an arbitrary continuous real-valued function. These conditions guaranteethe existence of a solution of problem (1) in L (cid:0) , T ; H (Ω) ∩ H (Ω) (cid:1) (see for instance [6–8]), and uniquenessand high order regularity can be deduced. The conditions (2)-(3) are at least satisfied by any polynomial ofodd degree with positive leading coefficient, and the matrix M is supposed to be constant only for the sake ofsimplicity. In fact, all our results remain true replacing the operator M ∆ by an elliptic operator L : Lu = − J X i,j =1 a i,j ( x ) u x i x j + J X i =1 b j ( x ) u x i + c ( x ) u, (4)where the coefficients a i,j , b i and c are smooth functions, and a i,j = a j,i (see, e.g., [8, p.292] for a definitionof elliptic operator). The analysis also remains true substituting the Dirichlet condition in (1) by Neumannconditions.The numerical analysis of reaction-diffusion equations takes advantage of many results available from thenumerical analysis of semi-linear parabolic partial differential equations (PDEs). The method of lines (MOL)is commonly used. By this method the PDE is first discretized in space by finite element or finite differencemethods, leading to a system of ordinary differential equations (ODEs). The resulting system of ODEs isthen discretized by fully implicit or implicit-explicit (IMEX) time-stepping methods (see for instance [9–16]).In [9–11], linear implicit-explicit multistep methods in time together with finite element methods in space areanalysed for a class of abstract semi-linear parabolic equations that includes a large class of reaction-diffusionsystems. The approaches in [9–11] are the same. The authors investigate approximate solutions expected to bein a tube around the exact solution. They proceeded by induction by adapting the time step k and the space step h and established that if k and k − h r , r ≥
2, are small enough then the global error of the scheme is of order p ( p = 1 , , ...,
5) in time and r in space. IMEX schemes with finite difference in space and Runge-Kutta of order1 and 2 in time are also analysed in [17, 18] for a class of reaction-diffusion systems. Otherwise, in [12, 13, 19]fully implicit numerical methods for reaction-diffusion equations with restrictive conditions on the nonlinearterm are introduced, combining finite elements in space and backward Euler, Crank-Nicolson or fractional-step θ methods in time. The resulting schemes are unconditionally stable (the time step is independent from thespace step) with order 1 or 2 of accuracy in time. The time-stepping method in [16] is constructed via a deferredcorrection strategy applied to the trapezoidal rule and is of arbitrary high order. However, this method concernsonly linear initial value problems (IVP) (resulting eventually from a MOL) satisfying a monotonicity conditionand has an issue for the starting procedure. Furthermore, the stability analysis proposed in this paper doesnot guarantee unconditional stability and/or an optimal a priori error estimate, when a full discretization isconsidered.In practice, the space-discretization of time-evolution PDEs leads to a stiff IVP of large dimension (werecall that a stiff problem is a problem extremely hard to solve by standard explicit step-by-step methods (see,e.g., [20]). To avoid overly small time steps, accurate approximate solutions for these IVPs require high ordertime-stepping methods having good stability properties (A-stable methods are of great interest). Backwarddifferentiation formulae (BDF) of order 1 and 2 are commonly used according to their A-stability. However,BDF methods of order 3 and higher lack stability properties (e.g. for systems with complex eigenvalues).Moreover, Runge-Kutta methods applied to such IVPs have order of convergence reduced to 1 or 2 (see [21]),and are inefficient when the IVPs are stiffer.The aim of this paper is to apply the deferred correction (DC) method introduced in [22] for the semi-discretization in time of the problem (1). The deferred correction method consists in a successive perturbation ITLE WILL BE SET BY THE PUBLISHER p + 2 at the stage p = 1 , , · · · of the correction. The order of accuracy of the DC schemes is guaranteed by a deferred correction condition(DCC). Applying the DC method to (1), the main difficulty is to prove that the resulting schemes satisfy DCCup to a certain stage p of the correction so that we obtain a time semi-discrete approximate solution with order2 p + 2 of accuracy. To overcome this difficulty, we suppose that the exact solution u of (1) is stationary in a smalltime interval [0 , (2 p + 1) k ], where k is a maximal time step for the time semi-discretized schemes and satisfies k µ < µ is the constant introduced in (3)). The stationary hypothesis is a simple trick to simplify ourproof. Indeed, the DCC is proven without restrictive condition in the case of IVP (see [22]), but the difficultyin the case of PDEs is related to the presence of unbounded operator. Each semi-discretized scheme in timeleads to a nonlinear elliptic equation that is discretized using the Galerkin finite element method. It resultsan arbitrary high-order unconditionally stable methods for the numerical solution of problem (1). A numericalillustration using the bistable reaction-diffusion equation with the schemes of order 2, 4, 6, 8 and 10 in time isgiven.The paper is organized as follows. We recall some algebraic property of finite difference operators in section1. In section 2 we introduce the semi-discretized schemes in time and prove the existence of a solution. Theanalysis of convergence and order of accuracy of solutions for the semi-discretized schemes in time is done insection 3. The fully discretized schemes are presented and analysed in section 4, and numerical experiments arecarried in section 5. Finite difference operators
In this section we recall main results from finite difference (FD) approximations. Details and proofs for theseresults can be found in [23]. For a time step k >
0, we denote t n = nk and t n +1 / = ( n + 1 / k , for each integer n . This implies that t = 0. We consider the time steps k such that 0 = t < t < · · · < t N = T is a partition of[0 , T ], for a nonnegative integer N . The centered, forward and backward difference operators D , D + and D − ,respectively, related to k and applied to a function v from [0 , T ] into a Banach space X (with norm k · k X ), aredefined as follows: Dv ( t n +1 / ) = v ( t n +1 ) − v ( t n ) k ,D + v ( t n ) = v ( t n +1 ) − v ( t n ) k , and D − v ( t n ) = v ( t n ) − v ( t n − ) k . The average operator is denoted by E : Ev ( t n +1 / ) = b v ( t n +1 ) = v ( t n +1 ) + v ( t n )2 . The composites of D + and D − are defined recursively. They commute, that is ( D + D − ) v ( t n ) = ( D − D + ) v ( t n ) = D − D + v ( t n ), and satisfy the identities( D + D − ) m v ( t n ) = k − m m X i =0 ( − i (cid:18) mi (cid:19) v ( t n + m − i ) , (5)and D − ( D + D − ) m v ( t n ) = k − m − m +1 X i =0 ( − i (cid:18) m + 1 i (cid:19) v ( t n + m − i ) , (6) TITLE WILL BE SET BY THE PUBLISHER for each integer m ≥ ≤ t n − m − ≤ t n + m ≤ T . If { v n } n is a sequence of approximation of v at thediscrete points t n , the finite difference operators apply to { v n } and we define Dv n +1 / = D + v n = D − v n +1 = v n +1 − v n k . and Ev n +1 / = b v n +1 = v n +1 + v n . We have the following three results:
Result 1
For nonnegative integers m and m , provided v ∈ C m + m ([0 , T ] , X ) and m ≤ n ≤ N − m , we have (cid:13)(cid:13) D m + D m − v ( t n ) (cid:13)(cid:13) ≤ max t n − m ≤ t ≤ t n + m (cid:13)(cid:13)(cid:13)(cid:13) d m + m vdt m + m ( t ) (cid:13)(cid:13)(cid:13)(cid:13) . (7) Result 2 (Central finite difference approximations)
There exists a sequences { c i } i ≥ of real numbers such that, for all v ∈ C p +3 ([0 , T ] , X ), where p is a positiveinteger, and p ≤ n ≤ N − − p , we have v ′ ( t n +1 / ) = v ( t n +1 ) − v ( t n ) k − p X i =1 c i +1 k i D ( D + D − ) i v ( t n +1 / ) + O ( k p +2 ) , (8)and v ( t n +1 / ) = v ( t n +1 ) + v ( t n )2 − p X i =1 c i k i ( D + D − ) i Ev ( t n +1 / ) + O ( k p +2 ) . (9)Table 1 gives the ten first coefficients c i . Table 1.
Ten first coefficients of central difference approximations (8) and (9) c c c c c c c c c c − − − − Result 3 (Interior central finite difference approximations)
For each positive integer p there exists reals c p , c p , · · · , c p p +1 such that, for each v ∈ C p +3 ([ a, b ] , X ) and auniform partition a = τ < τ < ... < τ p +1 = b of the interval [ a, b ], with τ n = a + nk , k = ( b − a ) / (2 p + 1) and τ p +1 / = ( a + b ) /
2, we have u ′ ( τ p +1 / ) = u ( b ) − u ( a ) b − a − b − a p X i =1 c p i +1 k i +1 D ( D + D − ) i u ( τ p +1 / ) + O ( k p +2 ) , (10)and u ( τ p +1 / ) = u ( b ) + u ( a )2 − p X i =1 c p i k i ( D + D − ) i Eu ( τ p +1 / ) + O ( k p +2 ) . (11)Table 2 gives the coefficients c pi for p = 1 , , , ITLE WILL BE SET BY THE PUBLISHER Table 2.
Coefficients of the approximations (10)-(11) for p = 1 , , , p c p c p c p c p c p c p c p c p
98 98
258 12524 125128 125128
498 34324 637128 133771920 10291024 10291024
818 2438 1917128 17253640 71731024 645577168 3273332768 3273332768 Semi-discrete schemes in time: existence of a solution
Hereafter we suppose that (1) has a unique solution u ∈ C p +4 (cid:0) [0 , T ] , H r +1 (Ω) ∩ H (Ω) (cid:1) , for some positiveintegers p and r . We denote by ( · , · ) the inner product in L (Ω) and by k ·k the corresponding norm. The norm inthe Sobolev spaces H m (Ω) will be noted k · k m , for each nonnegative integer m , and we note k · k ∞ = k · k L ∞ (Ω) .We use h and k to denote stepsizes for space and time discretizations, respectively. The letter C will denoteany constant independent from h and k , and that can be calculated explicitly in term of known quantities. Theexact value of C may change from a line to another line.As in [22], we can apply deferred correction method to (1) and deduce the following schemes:For j = 0, we have the implicit midpoint rule u ,n +1 − u ,n k − M ∆ (cid:18) u ,n +1 + u ,n (cid:19) + f (cid:18) u ,n +1 + u ,n (cid:19) = s ( t n +1 / ) , in Ω ,u ,n = 0 on ∂ Ω ,u , = u . (12)For j ≥
1, we have u j +2 ,n +1 − u j +2 ,n k − D Λ j u j,n +1 / − M ∆ (cid:16)b u j +2 ,n +1 − Γ j Eu j,n +1 / (cid:17) + f (cid:18) u j +2 ,n +1 + u j +2 ,n − Γ j Eu j,n +1 / (cid:19) = s ( t n +1 / ) , in Ω , for n ≥ j + 1 ,u j +2 ,n = 0 on ∂ Ω ,u j +2 , = u , (13)where Γ and Λ are finite differences operators defined for each positive integer j , and n ≥ j , byΛ j u ( t n ) = j X i =1 c i +1 k i ( D + D − ) i u ( t n ) = j X i =1 2 i X l =0 c i +1 ( − l (cid:18) il (cid:19) u ( t n + i − l ) , (14)and Γ j u ( t n ) = j X i =1 c i k i ( D + D − ) i u ( t n ) = j X i =1 2 i X l =0 c i ( − l (cid:18) il (cid:19) u ( t n + i − l ) . (15)The scheme (12) has unknowns (cid:8) u ,n (cid:9) Nn =1 corresponding to approximations of u ( t n ), expected to be of order2 of accuracy. For (13) the unknowns are (cid:8) u j +2 ,n (cid:9) Nn = j +1 , expected to be of order 2 j + 2, while (cid:8) u j,n (cid:9) Nn = j is supposed known from the preceding stage. To avoid computing approximate solution of (1) for t <
0, the
TITLE WILL BE SET BY THE PUBLISHER scheme (13) is used only for n ≥ j . For the starting values, 0 ≤ n ≤ j −
1, we consider the scheme Du j +2 ,n +1 / − j + 1 ¯Λ j D ¯ u j,n j +1 / − M ∆ (cid:16)b u j +2 ,n +1 − ¯Γ j E ¯ u j,n j +1 / (cid:17) + f (cid:16)b u j +2 ,n +1 − ¯Γ j E ¯ u j,n j +1 / (cid:17) = s ( t n +1 / ) ,u j +2 ,n = 0 on ∂ Ω ,u j +2 , = u , (16)where we set n j = (2 j + 1) n + j ,12 j + 1 ¯Λ j D ¯ u j, (2 j +1) n + j +1 / = k − j X i =1 2 i +1 X l =0 c j i +1 ( − l (cid:18) i + 1 l (cid:19) ¯ u j, (2 j +1) n + j + i − l +1 , (17)and ¯Γ j ¯ u j, (2 j +1) n + j = j X i =1 2 i X l =0 c j i ( − l (cid:18) il (cid:19) ¯ u j, (2 j +1) n + j + i − l . (18)This scheme is built from (10) and (11), for a = t n and b = t n +1 . (cid:8) ¯ u ,n (cid:9) Nn =1 is computed from (12) with timethe step k/ k . Similarly, (cid:8) ¯ u j,n (cid:9) Nn = j , j ≥
2, is computed from the scheme (13) with the time step k/ (2 j + 1) instead of k .To prove the existence of a solution for the schemes (12) and (13), we need the following lemma. Lemma 1.
Let k > such that k | τ (0) | ≤ / , and v ∈ L (Ω) . Then the elliptic problem u − kM ∆ u + kf ( u ) = v in Ω , (19) u = 0 on ∂ Ω , (20) has a solution u ∈ H (Ω) ∩ H (Ω) satisfying the inequality k u k ≤ C ( ||| M ||| /γ ) (cid:16) k − k v − u k + p γµ k∇ u k (cid:17) , (21) where γ is the smallest eigenvalue of the positive definite matrix M , ||| M ||| is any norm of the matrix M , andthe function τ and the scalar µ are defined in (2) and (3), respectively.Proof. The existence can be deduced from the Schaefer fixed point theorem [8, p. 504]. In fact, given u ∈ H (Ω) ∩ H (Ω), the problem w − kM ∆ w + kf ( u ) = v in Ω , (22) w = 0 on ∂ Ω , (23)has a unique solution w ∈ H (Ω) ∩ H (Ω) (see [8, p.317]). Consider the nonlinear mapping A : H (Ω) ∩ H (Ω) −→ H (Ω) ∩ H (Ω) , which maps u ∈ H (Ω) ∩ H (Ω) to the unique solution w = A [ u ] of (22)-(23). It is enough to prove that A iscontinuous, compact, and that the setΣ = (cid:8) u ∈ H (Ω) ∩ H (Ω) | u = λA [ u ] , for some λ ∈ [0 , (cid:9) (24) ITLE WILL BE SET BY THE PUBLISHER A is continuous. Indeed, let { u m } ∞ m =1 in H (Ω) ∩ H (Ω) which converges to u ∈ H (Ω) ∩ H (Ω).For each m = 1 , , · · · , let w m = A [ u m ] and w = A [ u ]. Then w − w m belongs to H (Ω) ∩ H (Ω) and satisfies theequation ( w − w m ) − kM ∆( w − w m ) + k ( f ( u ) − f ( u m )) = 0 in Ω . (25)The inner product of the last identity with w − w m yields k w − w m k + γk k∇ ( w − w m ) k + k ( f ( u ) − f ( u m ) , w − w m ) ≤ . (26)We can write, f ( u ( x )) − f ( u m ( x )) = Z df ( u ( x ) − ξ ( u ( x ) − u m ( x )))( u ( x ) − u m ( x )) dξ. Since u m −→ u in H (Ω) and H (Ω) ֒ → C (Ω), there exists a positive integer m such that m ≥ m impliesmax x ∈ Ω | u ( x ) − u m ( x ) | ≤ c k u − u m k ≤ , (27)where c is the constant from the Sobolev embedding. It follows that | f ( u ( x )) − f ( u m ( x )) | ≤ β | u ( x ) − u m ( x ) | , (28)where β = max | y |≤ c k u k | df ( y ) | . Therefore, by Cauchy-Schwartz inequality we have k | ( f ( u ) − f ( u m ) , w − w m ) | ≤ kβ k u − u m kk w − w m k ≤ ( kβ ) k u − u m k + 12 k w − w m k . The last inequality substituted into (26) yields k w − w m k + 2 γk k∇ ( w − w m ) k ≤ ( kβ ) k u − u m k . It follows that w m → w in H (Ω) when m → + ∞ . On the other hand, elliptic regularity results applied to theidentity (25) yields, owing to (28) and the last inequality, k w − w m k ≤ C (cid:0) k − k w − w m k + k f ( u ) − f ( u m ) k (cid:1) ≤ βC k u − u m k → m → + ∞ . Whence { w m } + ∞ m =1 converges to w in H (Ω) ∩ H (Ω), and the continuity of the mapping A follows.(ii) The mapping A is compact. Indeed, given a bounded sequence { u m } m ∈ N in H (Ω) ∩ H (Ω), from thecompact embedding H (Ω) ֒ → H (Ω) we can extract a subsequence (cid:8) u m j (cid:9) j ∈ N that converges to u strongly in H (Ω) and weakly in H (Ω). The subsequence (cid:8) u m j (cid:9) j ∈ N is then bounded in H (Ω) ∩ H (Ω). Let κ = sup m ∈ N k u m k and β ′ = max | y |≤ c ( κ + k u k ) | df ( y ) | . Therefore, proceeding exactly as in part (i), substituting m by m j , the inequality (27) bymax x ∈ Ω | u m j ( x ) | ≤ c sup m ∈ N k u m j k = c κ, TITLE WILL BE SET BY THE PUBLISHER and β by β ′ in (28), we deduce that w m j = A [ u m j ] → w strongly in H (Ω) ∩ H (Ω). Hence A is compact.(iii) The set Σ is bounded.Let u ∈ H (Ω) ∩ H (Ω) such that u = λA [ u ] for some λ ∈ (0 , u satisfies u − kM ∆ u + λkf ( u ) = λv in Ω , (29) u = 0 on ∂ Ω . (30)By elliptic regularity results we have k u k ≤ C k k − ( λv − u ) − λf ( u ) k = C k M ∆ u k . (31)The inner product of (29) with u , taking the boundary condition (30) into account, yields k u k + γk k∇ u k + λk Z Ω f ( u ) · udx = λ Z Ω v · udx. Without loss of generality we suppose that f (0) = 0, otherwise we change f by ˜ f = f − f (0) and v by ˜ v = v − kf (0).Then the monotonicity condition (2) combined with the hypothesis of the lemma yields λk Z Ω f ( u ) · udx ≥ αλk k u k q + λkτ (0) k u k ≥ αλk k u k q − k u k , ∀ λ ∈ (0 , . (32)From Cauchy-Schwartz inequality and the Cauchy inequality with ε = 1, we have λ Z Ω v · udx ≤ λ k v k + 14 k u k . Substituting the last two inequalities in the previous identity, we deduce that k u k + 2 γk k∇ u k ≤ λ k v k . (33)On the other hand, the inner product of (29) with − ∆ u yields γ k ∆ u k ≤ k − Z Ω ( λv − u ) · ( − ∆ u ) dx + Z Ω λf ( u ) · ∆ udx. (34)We can write f ( u ) · ∆ u = J X i =1 ∇ · ( f i ( u ) ∇ u i ) − J X i =1 (cid:18) df ( u ) (cid:18) ∂u∂x i (cid:19)(cid:19) · ∂u∂x i , and deduce from (3), the boundary condition and the hypothesis f (0) = 0 that Z Ω f ( u ) · ∆ udx = − J X i =1 Z Ω (cid:18) df ( u ) (cid:18) ∂u∂x i (cid:19)(cid:19) · ∂u∂x i dx ≤ µ J X i =1 (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂x i (cid:13)(cid:13)(cid:13)(cid:13) = µ k∇ u k . By Cauchy-Schwartz inequality and the Cauchy inequality with ε = 1 / (2 γ ) we have (cid:12)(cid:12)(cid:12)(cid:12) k − Z Ω ( λv − u ) · ( − ∆ u ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k − k λv − u kk ∆ u k ≤ γk k λv − u k + γ k ∆ u k . ITLE WILL BE SET BY THE PUBLISHER γ k ∆ u k ≤ k − k λv − u k + 2 λγµ k∇ u k . Therefore, k M ∆ u k ≤ ( ||| M ||| /γ ) (cid:0) k − k λv − u k + 2 γµ k∇ u k (cid:1) since 0 ≤ λ ≤
1, and we deduce from (31) that k u k ≤ C ( ||| M ||| /γ ) (cid:16) k − k λv − u k + p γµ k∇ u k (cid:17) . (35)The last inequality together with (33) yields k u k ≤ C ( ||| M ||| /γ ) k − (cid:16) √ p kµ (cid:17) k v k , and it follows that Σ is bounded. From (i)-(iii) we deduce by the Schaefer fixed point theorem that (19)-(20)has a solution u ∈ H (Ω) ∩ H (Ω) and (21) follows, taking λ = 1 in (35). (cid:3) The following theorem shows the existence of a solution for the schemes (12) and (13).
Theorem 1.
Suppose that u ∈ H (Ω) ∩ H (Ω) . Then, for each nonnegative integer n , the scheme (12) and(13) has a solution in H (Ω) ∩ H (Ω) .Proof. Proceeding by induction, the proof is immediate from Lemma 1 for a suitable choice of the functions u and v . For example, multiplying the first equation in (12) by k/
2, we deduce (19)-(20) for u = ( u ,n +1 + u ,n ) / v = ks ( t n +1 / ) / u ,n and k substituted by k/ (cid:3) Hereafter we suppose that u j,n ∈ H r +1 (Ω) ∩ H (Ω), for 1 ≤ j ≤ p + 1 and each n = 0 , · · · , N . Convergenceresults for these semi-discrete solutions are proven in section 3. Convergence and order of accuracy of the semi-discrete solution
The deferred correction condition (DCC) defined in [22] for ODEs applies to PDEs.
Definition 1.
Let u be the exact solution of (1). For a positive integer j , a sequence (cid:8) u j,n (cid:9) n ⊂ H (Ω) ofapproximations of u on the uniform partition t < t < · · · < t N = T , t n = nk , is said to satisfy the DeferredCorrection Condition (DCC) for the implicit midpoint rule if (cid:8) u j,n (cid:9) n approximates u ( t n ) with order j ofaccuracy in time, and for n = 1 , , ..., N − we have k ( D + D − ) D ( u j,n +1 / − u ( t n +1 / )) k + k D + D − ( u j,n +1 − u ( t n +1 )) k ≤ Ck j , (36) for each time steps k ≤ k , where k > is fixed and C is a constant independent from k . Remark 1.
Condition (36) is equivalent to (cid:13)(cid:13) Γ j (cid:0) u j,n − u ( t n ) (cid:1)(cid:13)(cid:13) ≤ Ck j +2 , (37) and (cid:13)(cid:13)(cid:13) (Λ j − Γ j ) D (cid:16) u j,n +1 / − u ( t n +1 / ) (cid:17)(cid:13)(cid:13)(cid:13) ≤ Ck j +2 , (38) for n = j, j + 1 , · · · , N − j . This is due to the transforms k i ( D + D − ) i (cid:0) u j,n − u ( t n ) (cid:1) = k i − X l =0 ( − l (cid:18) i − l (cid:19) D + D − (cid:0) u j,n − u ( t n ) (cid:1) , TITLE WILL BE SET BY THE PUBLISHER and k i ( D + D − ) i D (cid:16) u j,n +1 / − u ( t n +1 / ) (cid:17) = k i − X l =0 ( − l (cid:18) i − l (cid:19) ( D + D − ) D (cid:16) u j,n +1 / − u ( t n +1 / ) (cid:17) . The following theorem gives a sufficient condition for the semi-discrete schemes in time to converge with theexpected order of accuracy.
Theorem 2.
Let j be a positive integer and (cid:8) u j,n (cid:9) n ⊂ H (Ω) a sequence of approximations of u , on thediscrete points t = 0 < t < · · · < t N = T , satisfying DCC for the implicit midpoint rule. Suppose that k < k ,and that u j +2 , , ..., u j +2 ,j are given and satisfy k u j +2 ,n − u ( t n ) k ≤ Ck j +2 , for n = 0 , , ..., j. (39) Then the sequence (cid:8) u j +2 ,n (cid:9) n ≥ j , solution of the scheme (13) built from (cid:8) u j,n (cid:9) n , approximates u with order j + 2 of accuracy in time, and we have, for n = 0 , , · · · , N , k u j +2 ,n − u ( t n ) k + γk n X i = j k∇ b Θ j +2 ,i k ≤ Ck j +2 , (40) where Θ j +2 ,n = (cid:0) u j +2 ,n − u ( t n ) (cid:1) − Γ j (cid:0) u j,n − u ( t n ) (cid:1) , (41) and C is a constant depending only on j , T , M , u ∈ C j +3 (cid:0) [0 , T ] , H (Ω) (cid:1) , a Lipschitz constant on f and theDCC constant.Proof. Combining (13) and (1), we obtain the identity D Θ j +2 ,n +1 / + f (cid:0)b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) − f (cid:0)b u ( t n +1 ) − Γ j b u ( t n +1 ) (cid:1) − M ∆ b Θ j +2 ,n +1 = σ j +2 ,n +1 / + (Λ j − Γ j ) D ( u j,n +1 / − u ( t n +1 / )) , (42)where σ j +2 ,n +1 / = u ′ ( t n +1 / ) − Du ( t n +1 / ) + Λ j Du ( t n +1 / ) + f ( u ( t n +1 / )) − f (cid:0)b u ( t n +1 ) − Γ j b u ( t n +1 / ) (cid:1) − M ∆ (cid:0) u ( t n +1 / ) − b u ( t n +1 ) + Γ j b u ( t n +1 / ) (cid:1) . The inner product of (42) with b Θ j +2 ,n +1 , taking into account the monotonicity condition (2) and the fact that b Θ j +2 ,n +1 = 0 on ∂ Ω, yields( D Θ j +2 ,n + , b Θ j +2 ,n +1 ) + γ k∇ b Θ j +2 ,n +1 k ≤ τ ( b u ( t n +1 ) − Γ j b u ( t n +1 )) k b Θ j +2 ,n +1 k + (cid:16) σ j +2 ,n +1 / + (Λ j − Γ j ) D ( u j,n +1 / − u ( t n +1 / )) , b Θ j +2 ,n +1 (cid:17) . (43)From the central finite differences (8)-(9) and the mean value theorem we have k σ j +2 ,n +1 / k ≤ Ck j +2 , where C is a constant depending only on a Lipschitz condition on f and the norm of u as element of C j +3 (cid:0) [0 , T ] , H (Ω) (cid:1) ,and there exists 0 < k ≤ k such that k ≤ k implies that k b u ( t n +1 ) − Γ j b u ( t n +1 ) k ∞ ≤ k u ( t n +1 / ) − b u ( t n +1 ) + Γ j b u ( t n +1 ) k ∞ + k u ( t n +1 / ) k ∞ ≤ k u k L ∞ ( Q T ) , ITLE WILL BE SET BY THE PUBLISHER Q T = Ω × (0 , T ). It follows that, for k ≤ k , (cid:13)(cid:13) τ ( b u ( t n +1 ) − Γ j b u ( t n +1 )) (cid:13)(cid:13) ∞ ≤ max | y |≤ k u k L ∞ ( QT ) | τ ( y ) | =: µ. On the other hand, from the DCC we immediately have k (Λ j − Γ j ) D ( u j,n +1 / − u ( t n +1 / ) k ≤ Ck j +2 . Substituting the last inequalities in (43), taking into account the identity (cid:16) D Θ j +2 ,n +1 / , b Θ j +2 ,n +1 (cid:17) = 12 k (cid:0) k Θ j +2 ,n +1 k − k Θ j +2 ,n k (cid:1) , we deduce that k Θ j +2 ,n +1 k − k Θ j +2 ,n k + 2 kγ k∇ b Θ j +2 ,n +1 k ≤ Ck j +3 k b Θ j +2 ,n +1 k + 2 kµ k b Θ j +2 ,n +1 k . (44)This inequality yields k Θ j +2 ,n +1 k − k Θ j +2 ,n k ≤ Ck j +3 k b Θ j +2 ,n +1 k + 2 kµ k b Θ j +2 ,n +1 k , and, for µk <
2, we deduce from the inequality (cid:13)(cid:13)(cid:13) b Θ j +2 ,n +1 (cid:13)(cid:13)(cid:13) ≤ (cid:0)(cid:13)(cid:13) Θ j +2 ,n +1 (cid:13)(cid:13) + (cid:13)(cid:13) Θ j +2 ,n (cid:13)(cid:13)(cid:1) that (cid:13)(cid:13) Θ j +2 ,n +1 (cid:13)(cid:13) ≤ C k j +3 − µk + 2 + µk − µk (cid:13)(cid:13) Θ j +2 ,n (cid:13)(cid:13) . It follows by induction on n that (cid:13)(cid:13) Θ j +2 ,n (cid:13)(cid:13) ≤ C − µk (cid:18) µk − µk (cid:19) n − j − k j +2 + (cid:18) µk − µk (cid:19) n − j (cid:13)(cid:13) Θ j +2 ,j (cid:13)(cid:13) . From the hypothesis (39) and the DCC we have (cid:13)(cid:13) Θ j +2 ,j (cid:13)(cid:13) ≤ (cid:13)(cid:13) u j +2 ,j − u ( t j ) (cid:13)(cid:13) + (cid:13)(cid:13) Γ j (cid:0) u j,j − u ( t j ) (cid:1)(cid:13)(cid:13) ≤ Ck j +2 , (45)where C is a constant independent from k . Moreover, the sequence n(cid:16) µk − µk (cid:17) n o n is bounded above byexp(2 µT / (2 − ε )), for 0 ≤ µk ≤ ε <
2. Whence k Θ j +2 ,n k ≤ Ck j +2 . (46)Finally, by the triangle inequality, the identity (41) and the DCC, we have k u j +2 ,n − u ( t n ) k ≤ Ck j +2 + (cid:13)(cid:13) Γ j ( u j,n − u ( t n )) (cid:13)(cid:13) ≤ Ck j +2 , (47)where C is a constant depending only on j , T , µ , M , a Lipschitz constant on f and u as element of C j +3 (cid:0) [0 , T ] , H (Ω) (cid:1) .Substituting (46) in (44), we have k Θ j +2 ,n +1 k − k Θ j +2 ,n k + 2 kγ k∇ b Θ j +2 ,n +1 k ≤ Ck j +5 , TITLE WILL BE SET BY THE PUBLISHER and it follows by induction, taking (45) into account, that k Θ j +2 ,n +1 k + 2 kγ n X i = j k∇ b Θ j +2 ,i k ≤ Ck j +4 . Inequality (40) follows from (47) and the last inequality. (cid:3)
To prove DCC for the schemes (12) and (13) we need the following lemma:
Lemma 2.
The sequence (cid:8) u ,n (cid:9) n from the scheme (12) approximates u , the exact solution of (1), with order2 of accuracy. Furthermore, if u ( ., t ) = u for all t ∈ [0 , (2 p + 1) k ] , where k is the initial time step defined inthe introduction ( k µ < ), then we have k D − ( D + D − ) m Θ ,n +1 k + (cid:13)(cid:13) ( D + D − ) m Θ ,n +1 (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ( D + D − ) m b Θ ,n +1 (cid:13)(cid:13)(cid:13) + γk n X i = m k∇ ( D + D − ) m D b Θ ,i +1 / k ! / ≤ Ck , (48) for m = 0 , , , ..., p , n = m, m + 1 , · · · , N − m , and k ≤ k , where Θ ,n = u ,n − u ( t n ) , for n = 0 , , , · · · , N , µ is from (3), and C is a constant depending only on T , Ω , µ , k , M , the continuity of the source term S , thederivatives of f up to order m + 2 , and the derivatives of u with respect to the time variable t up to order m + 4 .Proof. According to Theorem 2, it is immediate that the sequence (cid:8) u ,n (cid:9) n from the scheme (12) approximates u with order 2 of accuracy in time, and k Θ ,n k + γk n X i =0 k∇ b Θ ,i k ≤ Ck , for n = 0 , , · · · , N, (49)where C is a constant depending only on T , Ω, a Lipschitz constant on f and the derivatives of u ∈ C (cid:0) [0 , T ] , H (Ω) (cid:1) .To prove (48) we proceed by induction on the integer m .1) The case m = 0.Combining (1) and (12), we obtain the identity D Θ ,n +1 / − M ∆ b Θ ,n +1 + h ( t n +1 ) = w ,n +1 / , (50)where h ( t n ) = f ( b u ,n ) − f ( b u ( t n )) = Z df ( K n ) ( b Θ ,n ) dτ , with K n = b u ( t n ) + τ b Θ ,n , and w ,n +1 / = (cid:2) u ′ ( t n +1 / ) − Du ( t n +1 / ) (cid:3) − M ∆ (cid:0) u ( t n +1 / ) − b u ( t n +1 ) (cid:1) − (cid:2) f ( u ( t n +1 / )) − f ( b u ( t n +1 )) (cid:3) . Applying D + to (50), we obtain DD + Θ ,n +1 / − M ∆ D + b Θ ,n +1 + D + h ( t n +1 ) = D + w ,n +1 / , ITLE WILL BE SET BY THE PUBLISHER D + b Θ ,n +1 yields k D + Θ ,n +1 k − k D + Θ ,n k + 2 γk k∇ D + b Θ ,n +1 k ≤ k (cid:16) − D + h ( t n +1 ) + D + w ,n +1 / , D + b Θ ,n +1 (cid:17) . (51)We can write D + h ( t n ) = Z df (cid:0) K n +11 (cid:1) ( D + b Θ ,n ) dτ + Z Z d f ( K n ) (cid:16) D + K n , b Θ ,n (cid:17) dτ dτ , (52)where, for n + i ≤ N , we have K ni +1 = K ni + τ i +1 ( K n +1 i − K ni ) = K n + i X l =1 X ≤ i < ···
0, depending only on T , Ω,the regularity of S , the first derivative of f , and the second derivative of u with respect to t , such that k K ni k ∞ ≤ R, for i = 1 , , · · · , p + 1 . (54)From the condition (3) we have (cid:16) df ( K n ) ( D + b Θ ,n ) , D + b Θ ,n (cid:17) ≥ − µ k D + b Θ ,n k . (55)From (54) and (7) we have, for almost every x ∈ Ω, (cid:12)(cid:12)(cid:12) d f ( K n ) (cid:16) D + K n , b Θ ,n +1 (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ max | y |≤ R (cid:12)(cid:12) d f ( y ) (cid:12)(cid:12) | D + K n ( x ) || b Θ ,n +1 ( x ) |≤ C (cid:16) | b Θ ,n +1 ( x ) | + | D + b Θ ,n +1 ( x ) || b Θ ,n +1 ( x ) | (cid:17) . Therefore, (cid:13)(cid:13)(cid:13) d f ( K n ) (cid:16) D + K n , b Θ ,n +1 (cid:17)(cid:13)(cid:13)(cid:13) ≤ C (cid:16) k b Θ ,n +1 k + k D + b Θ ,n +1 k L (Ω) k b Θ ,n +1 k L (Ω) (cid:17) , and we deduce from the Sobolev embedding H (Ω) ֒ → L (Ω) and the Poincar´e inequality that (cid:13)(cid:13)(cid:13) d f ( K n ) (cid:16) D + K n , b Θ ,n +1 (cid:17)(cid:13)(cid:13)(cid:13) ≤ C (cid:16) k b Θ ,n +1 k + k∇ D + b Θ ,n +1 kk∇ b Θ ,n +1 k (cid:17) . (56)This inequality and (55) together with the Cauchy-Schwartz inequality yield − k (cid:16) D + h ( t n +1 ) , D + b Θ ,n +1 (cid:17) ≤ kµ k D + b Θ ,n +1 k + 12 γk k∇ D + b Θ ,n +1 k + Ck k D + b Θ ,n +1 k (cid:16) k b Θ ,n +1 k + k∇ b Θ ,n +1 k k D + b Θ ,n +1 k (cid:17) , (57)4 TITLE WILL BE SET BY THE PUBLISHER where we have used the Cauchy inequality with ε = γ/ k∇ D + b Θ ,n +1 kk∇ b Θ ,n +1 kk D + b Θ ,n +1 k ≤ γ k∇ D + b Θ ,n +1 k + 12 γ k∇ b Θ ,n +1 k k D + b Θ ,n +1 k . According to (49), we have k∇ b Θ ,n +1 k k D + b Θ ,n +1 k ≤ k − k∇ b Θ ,n +1 k (cid:16) k b Θ ,n +2 k + k b Θ ,n +1 k (cid:17) ≤ Ck . (58)From Taylor’s formula with integral remainder we can write w ,n +1 / = k g ( t n +1 ) , where, according to (7), we have k D m + D m − g ( t n ) k ≤ C, for m ≤ n ≤ N − m , (59)for each nonnegative integers m and m such that m + m ≤ p + 1. C is a constant depending only on T ,the derivatives of f up to order m + m + 1, and the norm of u in C m + m +3 (cid:0) [0 , T ] , H (Ω) (cid:1) . It follows fromCauchy-Schwartz inequality that (cid:12)(cid:12)(cid:12)(cid:16) kD + w ,n +1 / , D + b Θ ,n +1 (cid:17)(cid:12)(cid:12)(cid:12) ≤ Ck k D + b Θ ,n +1 k . Substituting the last inequality and the inequality (57) in (51), taking (49) and (58) into account, we deducethat k D + Θ ,n +1 k −k D + Θ ,n k + γk k∇ D + b Θ ,n +1 k ≤ kµ k D + b Θ ,n +1 k + Ck k D + b Θ ,n +1 k , (60)where C is a constant depending only on T , Ω, S , the second derivative of f and u ∈ C ([0 , T ] , H (Ω)). Thisinequality yields k D + Θ ,n +1 k − k D + Θ ,n k ≤ kµ k D + b Θ ,n +1 k + Ck . Since kµ ≤ k µ <
2, it follows by induction that k D + Θ ,n k ≤ Ck (cid:18) kµ − kµ (cid:19) n + (cid:18) kµ − kµ (cid:19) n k D + Θ , k . The condition u ( t n ) = u , for 0 ≤ t n ≤ (2 p + 1) k , implies k D + Θ , k = 0. Whence k D − Θ ,n k = k D + Θ ,n − k ≤ Ck , for n = 1 , , · · · , N. (61)Substituting (61) in the right hand side of (60), we deduce that k D − Θ ,n k + γk n X l =0 k∇ D + b Θ ,l k ≤ Ck . (62)On the other hand, by the elliptic regularity results applied to (50), we deduce from (54), (59) for m = m = 0,and (61) that k b Θ ,n +1 k ≤ C (cid:16) k D − Θ ,n +1 k + k h ( t n +1 ) k + k w ,n +1 / k (cid:17) ≤ Ck . Inequality (48) for m = 0 holds from (49), (62) and the last inequality.2) Inequality (48) for m + 1, assuming that it holds for arbitrary m ≤ p − ITLE WILL BE SET BY THE PUBLISHER D + D − ) m +1 to the identity (50) and take the inner product of the resulting identity with( D + D − ) m +1 b Θ ,n +1 to obtain, as in (51), k ( D + D − ) m +1 Θ ,n +1 k − k ( D + D − ) m +1 Θ ,n k + 2 γk k∇ ( D + D − ) m +1 b Θ ,n +1 k ≤ k (cid:16) − ( D + D − ) m +1 h ( t n +1 ) + ( D + D − ) m +1 w ,n +1 / , ( D + D − ) m +1 b Θ ,n +1 (cid:17) . (63)As in [22] we can write D s + h ( t n ) = s +1 X i =1 X | α i | = s L n,si,α i , for s = 1 , , ..., p + 1 , and n ≤ N − s, (64)where α i = ( α i , · · · , α i − i , α ii ) ∈ { , , · · · , s } i − ×{ , , · · · , s − i + 1 } . L n,si,α i is a linear combination of the quantities L n,si,α i ,β i = Z [0 , i d i F ( K n + s +1 − ii ) (cid:18) D α i − i + K n + β i − i i − , · · · , D α i + K n + β i , D α ii + b Θ ,n + β ii (cid:19) dτ i , where β i = ( β i , · · · , β i − i , β ii ) ∈ { , , · · · , s } i − × { , , · · · , s − i + 1 } with β li + α li ≤ s − l + 1, for l = 1 , · · · , i , and dτ i = dτ · · · dτ i . From (54) and the regularity of f we have (cid:13)(cid:13) d i f ( K ni ) (cid:13)(cid:13) ∞ ≤ C i , for i = 1 , , ..., p + 1 , ≤ n ≤ N − i + 1 , (65)where C i is a constant depending only on T , the i-th derivative of f and the second derivative of u . From theinduction hypothesis (48), the Sobolev embedding H (Ω) ֒ → L ∞ (Ω), and inequality (7), we have k D l + K ni k ∞ ≤ C, for 1 ≤ l ≤ m + 2 , ≤ n ≤ N − i − l + 1 , (66)and k D l + b Θ ,n k ≤ Ck , for 1 ≤ l ≤ m + 1 , ≤ n ≤ N − l. (67)- For i = 1 we have L n,s ,α = Z df ( K n + s ) (cid:16) D s + b Θ ,n (cid:17) dτ, and, by taking s = 2 m + 2, it follows from (3) that (cid:16) L n − m, m +21 ,α , ( D + D − ) m +1 b Θ ,n +1 (cid:17) ≥ − µ k ( D + D − ) m +1 b Θ ,n +1 k (68)since D m +2+ b Θ ,n − m = ( D + D − ) m +1 b Θ ,n +1 . - For i = 2 and | α | ≤ m + 2, we have 1 ≤ α ≤ m + 2 and 0 ≤ α ≤ m + 1. It follows by the triangle inequality,the inequalities (7) and (65)-(67) that k L n,s ∗ ,α ,β k ≤ (cid:13)(cid:13)(cid:13) d f (cid:16) K n + s ∗ − (cid:17)(cid:13)(cid:13)(cid:13) ∞ k D α + K n + β k ∞ k D α + b Θ ,n k ≤ Ck , for s ∗ ≤ m + 2 . (69)- For i ≥ | α i | ≤ m + 3, we have 1 ≤ α li ≤ m + 2, for l = 1 , , · · · , i −
1, and 0 ≤ α ii ≤ m + 1. It follows bythe triangle inequality, the inequalities (7) and (65)-(67) that, for s ∗ ≤ m + 3, k L n,s ∗ i,α i ,β i k ≤ k d i f ( K n + s ∗ +1 − ii ) k ∞ k D α ii + b Θ ,n + β ii k i − Y l =1 k D α li + K n + β li l k ∞ ≤ Ck . (70)6 TITLE WILL BE SET BY THE PUBLISHER
From the identity (64), inequalities (68)-(70) yield (cid:16) − ( D + D − ) m +1 h ( t n +1 ) , ( D + D − ) m +1 b Θ ,n +1 (cid:17) ≤ µ k ( D + D − ) m +1 b Θ ,n +1 k + Ck k ( D + D − ) m +1 b Θ ,n +1 k . (71)From inequality (59) we have k ( D + D − ) m +1 w ,n +1 / k ≤ Ck . (72)Substituting (71) and (72) in (63), we obtain k ( D + D − ) m +1 Θ ,n +1 k − k ( D + D − ) m +1 Θ ,n k + 2 γk k∇ ( D + D − ) m +1 b Θ ,n +1 k ≤ kµ k ( D + D − ) m +1 b Θ ,n +1 k + Ck k ( D + D − ) m +1 b Θ ,n +1 k . (73)Proceeding as in (60), we deduce by induction that k ( D + D − ) m +1 Θ ,n k ≤ (cid:0) Ck + k ( D + D − ) m +1 Θ ,m +1 k (cid:1) (cid:18) kµ − kµ (cid:19) n − m − . Since u ( t n ) = u for 0 ≤ t n ≤ (2 p + 1) k , we have k ( D + D − ) m +1 Θ ,m +1 k = 0, for m ≤ p −
1. Whence k ( D + D − ) m +1 Θ ,n k ≤ Ck , for n = m + 1 , m + 2 , · · · , N − m − . (74)Substituting (74) in the right hand side of (73), we deduce by induction that k ( D + D − ) m +1 Θ ,n k + 2 γk n X i = m +1 k∇ ( D + D − ) m +1 b Θ ,i k ≤ Ck . It is immediate from (65)-(67) that k L n, m +11 ,α k ≤ k df ( K n +2 m +11 ) k ∞ k D m +1+ b Θ ,n k ≤ Ck . Therefore, applying D − ( D + D − ) m to (50), we deduce from the elliptic regularity inequality, the identity (64),the last inequality, the inequalities (69)-(70), (74) and (59) that k D − ( D + D − ) m b Θ ,n +1 k ≤ k D − ( D + D − ) m (cid:16) D Θ ,n +1 / + h ( t n +1 ) + w ,n +1 (cid:17) k ≤ Ck . It follows that k ( D + D − ) m +1 Θ ,n +1 k + γk n X i = m +1 k∇ ( D + D − ) m +1 b Θ ,i k ! / + k D − ( D + D − ) m b Θ ,n +1 k ≤ Ck . (75)Otherwise, applying D + ( D + D − ) m +1 to (50), the same reasoning, taking the induction hypothesis and theinequality (75) into account, yields (48) for m + 1. Finally, we deduce by induction that Lemma 2 is true foreach m = 0 , , · · · , p . (cid:3) The following theorem shows DCC for the schemes (12) and (13) .
Theorem 3.
Suppose that the exact solution u of (1) satisfies u ( ., t ) = u for each t ∈ [0 , (2 p + 1) k ] , where k > is a fixed real such that k µ < . Then, for k ≤ k , each sequence (cid:8) u j,n (cid:9) n , j = 1 , , ..., p + 1 , from the ITLE WILL BE SET BY THE PUBLISHER schemes (12) or (13) approximates u with order j of accuracy in time and we have the estimate (cid:13)(cid:13) ( D + D − ) m (cid:0)b u j,n +1 − b u ( t n +1 ) (cid:1)(cid:13)(cid:13) + vuut k n X i = m k∇ D − ( D + D − ) m ( b u j,i − b u ( t i )) k + k D − ( D + D − ) m (cid:0) u j,n +1 − u ( t n +1 ) (cid:1) k + (cid:13)(cid:13) ( D + D − ) m (cid:0) u j,n +1 − u ( t n +1 ) (cid:1)(cid:13)(cid:13) ≤ Ck j . (76) for m = 0 , , ..., p − j and n = m + j − , m + j, ..., N − j − m , where µ is from (3), and C is a constant dependingonly on m , T , µ , k , M , the function S , and the derivatives of f and u = u ( t ) up to order m + 2 j and m + 2 j + 2 , respectively.Proof. We proceed by induction on j = 1 , , ..., p + 1, and the case j = 1 results from Lemma 2. Suppose that (cid:8) u j,n (cid:9) n satisfies (76) up to an arbitrary order j ≤ p . Let us prove that the theorem is still true for j + 1.Since (cid:8) u j,n (cid:9) n satisfies (76), it also satisfies DCC, and then Theorem 2 together with the condition u ( ., t ) = u in [0 , (2 p + 1) k ] implies that (cid:8) u j +2 ,n (cid:9) n approximates u with order 2 j + 2 of accuracy in time. Therefore, it isenough to establish (76) for j + 1. We can rewrite the identity (42) as follows D Θ j +2 ,n +1 / − M ∆ b Θ j +2 ,n +1 + H ( t n +1 ) = w j +2 ,n +1 / , (77)where H ( t n +1 ) = Z df (cid:16)b u ( t n +1 ) − Γ j b u ( t n +1 ) + τ b Θ j +2 ,n +1 (cid:17) (cid:16) b Θ j +2 ,n +1 (cid:17) dτ , and w j +2 ,n +1 / = σ j +2 ,n +1 / + (Λ j − Γ j ) D ( u j,n +1 / − u ( t n +1 / )) . Here Θ j +2 ,n +1 and σ j +2 ,n +1 / are as in Theorem 2. From the central finite difference (8)-(9) and the regularityof u with respect to t , we can write σ j +2 ,n +1 / = k j +2 G ( t n +1 / ) , where k D m + D m − G ( t n ) k ≤ C, for m ≤ n ≤ N − m , for each nonnegative integers m and m such that m + m ≤ p − j + 1. C is a constant depending only on T , the derivatives of f up to order m + m + 2 j + 1 and the norm of u in C m + m +2 j +3 (cid:0) [0 , T ] , H (Ω) (cid:1) . Onthe other hand, from the induction hypothesis and Remark 1, we immediately have k D − ( D + D − ) m (Λ j − Γ j )( u j,n − u ( t n )) k ≤ Ck j +2 , for m = 0 , , ..., p − ( j + 1) . The last two inequalities implies that k D m + D m − w j +2 ,n +1 / k ≤ Ck j +2 , for m + m ≤ p − j − , and m + j ≤ n ≤ N − m − j −
1. Therefore, the reasoning from Lemma 2, substituting the functions h by H , w ,n +1 / by w j +2 ,n +1 / , b Θ ,n +1 by b Θ j +2 ,n +1 and k by k j +2 , yields k D − ( D + D − ) m Θ j +2 ,n +1 k + (cid:13)(cid:13) ( D + D − ) m Θ j +2 ,n +1 (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ( D + D − ) m b Θ j +2 ,n +1 (cid:13)(cid:13)(cid:13) + k n X i = m k∇ D ( D + D − ) m b Θ j +2 ,i +1 / k ! / ≤ Ck j +2 , TITLE WILL BE SET BY THE PUBLISHER for m = 0 , , ..., p − ( j + 1), and (76) for j + 1 follows by the triangle inequality. Inequality (76) then holds forarbitrary integer j ≤ p + 1. (cid:3) Fully discretized schemes and convergence results
Let S h be a finite dimensional subspace of H (Ω) and { φ i } N h i =1 a basis for S h consisting in continuous piecewisepolynomials of degree r ≥ S h ; the integer r is related to the regularity of the exact solution of (1) in space). We suppose that there exist an interpolatingoperator I rh from H (Ω) onto S h and a constant c > ≤ l ≤ r implies k v − I rh v k + h k∇ ( v − I rh v ) k ≤ ch l +1 | v | l +1 , , Ω , ∀ v ∈ H l +1 (Ω) ∩ H (Ω) , (78)and k v − I rh v k L (Ω) + h k∇ ( v − I rh v ) k L (Ω) ≤ ch l +1 | v | l +1 , , Ω , ∀ v ∈ W l +1 , (Ω) ∩ H (Ω) , (79)where | · | l +1 ,ρ, Ω is the following seminorm in W l +1 ,ρ (Ω): | v | l +1 ,ρ, Ω = X | α | = l +1 | ∂ α v | L ρ (Ω) . We say that S h satisfies the inverse inequality if k v h k ∞ ≤ ch m − d/ k v h k m , ∀ v h ∈ S h , and m = 0 , . (80)The estimates (78) and (79) hold when S h is obtained from a shape-regular family of meshes {T h } h> [24,Corollary 1.109 & 1.110 ] while (80) is due to [25, Theorem 3.2.6] or [24, Lemma 1.142] for a family of meshes {T h } h> that is shape-regular and quasi-uniform. We consider the elliptic operator R h , orthogonal projectionof H (Ω) onto S h with respect to the inner product ( v, w ) ( M ∇ v, ∇ w ). Proceeding as in [13, Theorem 1.1],we deduce from (78) that k R h v − v k + h k∇ ( R h v − v ) k ≤ ch l +1 k v k H l +1 (Ω) , ∀ v ∈ H (Ω) ∩ H l +1 (Ω) , ≤ l ≤ r. (81)Furthermore, if S h satisfies the inverse inequality (80), we deduce from (81) and (78) for l = 1, and (79) for l = 0 together with the continuous embedding H (Ω) ֒ → W , (Ω) ֒ → L ∞ (Ω), that k R h v k ∞ ≤ k R h v − I rh v k ∞ + k v − I rh v k ∞ + k v k ∞ ≤ ch / k v k + C k v k , (82)for each v ∈ H (Ω) ∩ H (Ω).For j = 0 , , , · · · , p and each positive integer n ≤ N , we look for a function u j +2 ,nh ∈ H (Ω) of the form u j +2 ,nh = N h X l =1 U j +2 ,nl φ l , (83)satisfying (cid:16) Du j +2 ,n +1 / h − Λ j Du j,n +1 / h , φ (cid:17) + (cid:16) M ∇ (cid:16) Eu j +2 ,n +1 / h − Γ j Eu j,n +1 / h (cid:17) , ∇ φ (cid:17) + (cid:16) f (cid:16) Eu j +2 ,n +1 / h − Γ j Eu j,n +1 / h (cid:17) , φ (cid:17) = (cid:0) s ( t n +1 / ) , φ (cid:1) , ∀ φ ∈ S h , and n ≥ j (84) u j +2 , h = R h u , (85) ITLE WILL BE SET BY THE PUBLISHER j Du j,n +1 / h = Γ j b u j,n +1 / h = 0 if j = 0. The scheme (84)-(85), denoted DC(2j+2), constitutes a fulldiscretization of the problem (1) with deferred correction in time, at the discrete points 0 = t < t < · · · < t N = T , t n = nk , and finite element in space. For the starting values in (84)-(85), 0 ≤ n ≤ j − (cid:18) Du j +2 ,n +1 / h − j + 1 ¯Λ j D ¯ u j,n j +1 / h + f ( b u j +2 ,n +1 / h − ¯Γ j E ¯ u j,n j +1 / h ) , φ (cid:19) + (cid:16) M ∇ (cid:16)b u j +2 ,n +1 / h − ¯Γ j E ¯ u j,n j +1 / h (cid:17) , ∇ φ (cid:17) = (cid:0) s ( t n +1 / ) , φ (cid:1) , ∀ φ ∈ S h , (86) u j +2 , h = R h u , (87)The following theorem proves the existence of a solution for the schemes (84)-(85). Theorem 4 (Existence of a solution for the fully discretized scheme) . We suppose that k | τ (0) | < . Then, foreach j = 1 , , · · · , there exists a sequence n u j,nh o Nn =0 of elements of the form (83) satisfying (84)-(85). To prove this theorem we need the following lemma which is an adaptation of the lemma on zeros of a vectorfield [8, p.493].
Lemma 3.
Let m be a positive integer and v : R m → R m a continuous function satisfying v ( z ) · z ≥ if k z k ∗ = R, (88) for a positive real R , where k . k ∗ is an arbitrary norm on R m . Then there exists a point z in the closed ball B (0 , R ) = { z ∈ R m : k z k ∗ ≤ R } such that v ( z ) = 0 . Proof of Lemma 3.
Suppose that v ( z ) = 0 for each z ∈ B (0 , R ). The mapping ϕ : B (0 , R ) → B (0 , R )defined by ϕ ( z ) = − R k v ( z ) k ∗ v ( z )is continuous. Since B (0 , R ) is a compact and convex subset of R m , we deduce from Schauder’s fixed-pointtheorem [8, p.502] that ϕ has a fixed point z ∈ B (0 , R ). Therefore, k z k ∗ = R , and this leads to the contradiction0 < | z | = ϕ ( z ) · z = − R k v ( z ) k ∗ v ( z ) · z ≤ . (cid:3) Proof of Theorem 4.
We proceed by double induction on j = 1 , , · · · and n = 0 , , · · · , N , using Lemma 3 for thefunction v : R N h → R N h defined by v l ( z ) = (cid:18) z h − a h k , φ l (cid:19) + ( M ∇ z h , ∇ φ l ) + (cid:0) f ( z h ) − s ( t n +1 / ) , φ l (cid:1) , (89)0 TITLE WILL BE SET BY THE PUBLISHER for l = 1 , · · · , N h , where a h ∈ S h is fixed and z h is the unique element of S h associated to z ∈ R N h and definedby z h = N h X l =1 z l φ l . We take k z k ∗ = k z h k . The function v is continuous. For j = 1, we have u , h = R h u and, supposing that u ,nh exists for an arbitrary integer n < N and taking a h = u ,nh in (89), we have v ( z ) · z = z h − u ,nh k , z h ! + ( M ∇ z h , ∇ z h ) + (cid:0) f ( z h ) − s ( t n +1 / ) , z h (cid:1) ≥ k z h k k h (2 + kτ (0)) k z h k − k u ,nh k − k (cid:0) k f (0) k + k s ( t n +1 / ) k (cid:1)i ≥ , (90)for k z k ∗ = 12 + kτ (0) (cid:16) k u ,nh k + k k s ( t n +1 / ) k + k k f (0) k (cid:17) := R. Then, from Lemma 3, there exists a point z in the closed ball B (0 , R ) of (cid:0) R N h , k · k ∗ (cid:1) such that v ( z ) = 0. Taking U ,n +1 = (cid:16) U ,n +11 , · · · , U ,n +1 N h (cid:17) = 2 z − U ,n , we have v (cid:18) U ,n +1 + U ,n (cid:19) · e l = 0 , for each e l in the standard basis of R N h . The last identity implies the existence of u ,n +1 h of the form (83)satisfying (84)-(85). Moreover, if n u j,nh o Nn =0 exists and satisfies (84)-(85), for an arbitrary integer j ≥
1, thenwe have u j +2 , h = R h u , and the existence of u j +2 ,n +1 h is immediate from the existence of u j +2 ,nh , proceedingas in the case j = 1, taking a h = u j +2 ,nh − Γ j b u j,n +1 + 0 . k Λ j Du j,n +1 / in (89). (cid:3) The following theorem shows the convergence and order of accuracy of the fully discretized schemes.
Theorem 5 (Order of convergence of the fully discretized schemes) . Suppose that the exact solution u of (1)is C p +4 (cid:0) [0 , T ] , H r +1 (Ω) ∩ H (Ω) (cid:1) and satisfies u ( ., t ) = u for t ∈ [0 , (2 p + 1) k ] , where p is a positive integerand k > is a real such that k max { µ , τ (0) } < , µ and τ are defined in (2)-(3). In addition, suppose that S h satisfies the inverse inequality (80). Then, for j = 1 , , · · · , p + 1 , the solution n u j,nh o Nn =0 of the scheme(84)-(85) approximates u with order j of accuracy in time and order r + 1 in space, that is k u j,nh − u ( t n ) k + h (cid:13)(cid:13)(cid:13) ∇ (cid:16) u j,nh − u ( t n ) (cid:17)(cid:13)(cid:13)(cid:13) ≤ C ( k j + h r +1 ) , (91) for k < k . Furthermore, we have the estimate k u j,nh − R h u j,n k + k n X i =0 k D ( u j,i +1 / h − R h u j,i +1 / ) k + 2 αk n X i =0 k u j,ih − R h u j,i k qL q (Ω) ≤ Ch r +2 , (92) where C is a constant depending only on j , T , Ω , M , k , µ and the derivatives of S , f and u . ITLE WILL BE SET BY THE PUBLISHER Proof.
Inequality (91) is immediate from (92) by quadruple triangle inequality, writing u j,nh − u ( t n ) = (cid:16) u j,nh − R h u j,n (cid:17) − [ u ( t n ) − u j,n ] − [ u ( t n ) − R h u ( t n )] + (cid:2) u ( t n ) − u j,n − R h ( u ( t n ) − u j,n ) (cid:3) , and taking (81) and (76) into account. Therefore, we just need to establish (92). We proceed by induction on j = 1 , , · · · , p + 1. For this purpose, we need the following claim which proof is a straightforward application ofthe mean value theorem, the triangle inequality, and inequalities (76), (81)-(82). Claim 1.
There exist < k ≤ k and h > such that k ≤ k and h ≤ h imply, k R h ( b u j +2 ,n +1 − Γ j b u j,n +1 ) k ∞ ≤ C k u k L ∞ (0 ,T,H (Ω)) , (93) and k w j +2 ,n +1 / h k ≤ Ch r +1 , (94) for each j = 0 , , · · · , p , and n = 0 , , · · · , N , where we define w j +2 ,n +1 / h = f (cid:0)b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) − f (cid:0) R h ( b u j +2 ,n +1 − Γ j b u j,n +1 ) (cid:1) + D (cid:16) u j +2 ,n +1 / − Λ j u j,n +1 / (cid:17) − R h D (cid:16) u j +2 ,n +1 / − Λ j u j,n +1 / (cid:17) , (95) and we set u ,n = 0 .
1. The case j = 1. We proceed in two steps:(i) First, we are going to prove the inequality k u ,nh − R h u ,n k + 2 γk n X i =0 k∇ E ( u ,i +1 / h − R h u ,i +1 / ) k + 2 αk n X i =0 k E ( u ,i +1 / h − R h u ,i +1 / ) k qL q (Ω) ≤ Ch r +2 . (96)The scheme (12) yields (cid:16) Du ,n +1 / , φ (cid:17) + (cid:0) M ∇ b u ,n +1 , ∇ φ (cid:1) + Z Ω f ( b u ,n +1 ) φdx = (cid:0) s ( t n +1 / ) , φ (cid:1) , ∀ φ ∈ S h . Therefore, combining this identity and (84), for j = 0, we deduce that (cid:16) D Θ ,n +1 / h , φ (cid:17) + (cid:16) M ∇ b Θ ,n +1 h , ∇ φ (cid:17) + Z Ω (cid:16) f ( b u ,n +1 h ) − f ( R h b u ,n +1 ) (cid:17) φdx = (cid:16) w ,n +1 / h , φ (cid:17) + (cid:0) M ∇ (cid:0)b u ,n +1 − R h b u ,n +1 (cid:1) , ∇ φ (cid:1) , ∀ φ ∈ S h , (97)where Θ ,nh = u ,nh − R h u ,n , and w ,n +1 / h is defined in (95). Hypothesis (2) and inequality (93) yield Z Ω (cid:16) f ( b u ,n +1 h ) − f ( R h b u ,n +1 ) (cid:17) b Θ ,n +1 h dx ≥ α k b Θ ,n +1 h k qL q (Ω) − µ k b Θ ,n +1 h k , (98)where µ = max | y |≤ k u k L ∞ ( ,T ; H ) | τ ( y ) | . TITLE WILL BE SET BY THE PUBLISHER
From the properties of orthogonal projection we have, (cid:0) M ∇ (cid:0)b u ,n +1 − R h b u ,n +1 (cid:1) , ∇ φ (cid:1) = 0 , ∀ φ ∈ S h . (99)Therefore, choosing φ = b Θ ,n +1 h in (97), we deduce from the Cauchy-Schwartz inequality and the inequalities(94) and (98) that (cid:16) D Θ ,n +1 / h , b Θ ,n +1 h (cid:17) + γ k∇ b Θ ,n +1 h k + α k b Θ ,n +1 h k qL q (Ω) ≤ Ch r +1 k b Θ ,n +1 h k + µ k b Θ ,n +1 h k , (100)for 0 < k ≤ k and 0 < h ≤ h . This inequality yields (cid:16) D Θ ,n +1 / h , b Θ ,n +1 h (cid:17) ≤ Ch r +1 (cid:13)(cid:13)(cid:13) b Θ ,n +1 h (cid:13)(cid:13)(cid:13) + µ (cid:13)(cid:13)(cid:13) b Θ ,n +1 h (cid:13)(cid:13)(cid:13) , and it follows for 0 < kµ ≤ k µ < (cid:13)(cid:13)(cid:13) Θ ,n +1 h (cid:13)(cid:13)(cid:13) ≤ C k − kµh r +1 + 2 + kµ − kµ (cid:13)(cid:13)(cid:13) Θ ,nh (cid:13)(cid:13)(cid:13) . Proceeding by induction as in Theorem 2, the last inequality yields (cid:13)(cid:13)(cid:13) Θ ,nh (cid:13)(cid:13)(cid:13) ≤ (cid:16) nkCh r +1 + (cid:13)(cid:13)(cid:13) Θ , h (cid:13)(cid:13)(cid:13)(cid:17) (cid:18) kµ − kµ (cid:19) n ≤ Ch r +1 (101)since nk ≤ T and Θ , h = 0. Inequality (96) follows by substituting (101) in (100).(ii) Now we are going to prove the inequality k n X i =0 (cid:13)(cid:13)(cid:13) D Θ ,n +1 / h (cid:13)(cid:13)(cid:13) + γ k∇ Θ ,n +1 h k ≤ Ch r +2 . (102)We choose φ = D Θ ,n +1 / h in (97) and obtain Z Ω (cid:16) f ( b u ,n +1 h ) − f ( R h b u ,n +1 ) (cid:17) D Θ ,n +1 / h dx + (cid:16) M ∇ b Θ ,n +1 h , ∇ D Θ ,n +1 / h (cid:17) + (cid:13)(cid:13)(cid:13) D Θ ,n +1 / h (cid:13)(cid:13)(cid:13) = (cid:16) w ,n +1 / h , D Θ ,n +1 / h (cid:17) . (103)We can write f ( b u ,n +1 h ) − f ( R h b u ,n +1 ) = Z df (cid:16) R h b u ,n +1 + ξ b Θ ,n +1 h (cid:17) (cid:16) b Θ ,n +1 h (cid:17) dξ. From the inverse inequality (80) and the inequality (101), we have k b Θ ,n +1 h k ∞ ≤ ch − / k Θ ,nh k ≤ Ch r − / , r ≥ . (104)This inequality together with (93) implies that there exists 0 < h ≤ h such that, for 0 < h ≤ h , we have k R h b u ,n +1 + ξ b Θ ,n +1 h k ∞ ≤ k u k L ∞ ( ,T ; H (Ω) ) . ITLE WILL BE SET BY THE PUBLISHER (cid:13)(cid:13)(cid:13) f ( b u ,n +1 h ) − f ( R h b u ,n +1 ) (cid:13)(cid:13)(cid:13) ≤ max | y |≤ k u k L ∞ ( ,T ; H ) | df ( y ) | (cid:13)(cid:13)(cid:13) b Θ ,n +1 h (cid:13)(cid:13)(cid:13) ≤ C (cid:13)(cid:13)(cid:13) b Θ ,n +1 h (cid:13)(cid:13)(cid:13) . (105)Substituting (105) in (103), we deduce by Cauchy-Schwartz inequality and (94) that k k D Θ ,n +1 / h k + (cid:16) M ∇ Θ ,n +1 h , ∇ Θ ,n +1 h (cid:17) − (cid:16) M ∇ Θ ,nh , ∇ Θ ,nh (cid:17) ≤ Ckh r +2 , for n = 0 , , · · · , N −
1. It follows the inequality k n X i =0 (cid:13)(cid:13)(cid:13) D Θ ,n +1 / h (cid:13)(cid:13)(cid:13) + (cid:16) M ∇ Θ ,n +1 h , ∇ Θ ,n +1 h (cid:17) ≤ Cnkh r +2 since Θ , h = 0. The last inequality gives exactly (102), where C is a constant depending only on T , Ω, k i +1 , h i , i = 1 ,
2, and the derivatives of f and u .Estimates (96) and (102) gives (92) for j = 1.2. Here we prove inequality (92) for j + 1, assuming that it holds up to order j , 1 ≤ j ≤ p .From the scheme (13) we have (cid:16) Du j +2 ,n +1 / − Λ j Du j,n +1 / , φ (cid:17) + (cid:0) M ∇ (cid:0)b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) , ∇ φ (cid:1) + Z Ω f (cid:0)b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) φdx = (cid:0) s ( t n +1 / ) , φ (cid:1) , ∀ φ ∈ S h . (106)Combining this identity and (84), we deduce that (cid:16) D Θ j +2 ,n +1 / h + f (cid:16)b u j +2 ,n +1 h − Γ j b u j,n +1 h (cid:17) − f (cid:0) R h ( b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) , φ (cid:17) + ( M ∇ b Θ j +2 ,n +1 h , ∇ φ ) = (cid:16) w j +2 ,n +1 / h + (Λ j − Γ j ) D ( u j,n +1 / h − R h u j,n +1 / ) , φ (cid:17) , (107)for any φ ∈ S h , where we defineΘ j +2 ,nh = u j +2 ,nh − R h u j +2 ,n − Γ j ( u j,nh − R h u j,n ) , and we use the identity (cid:0) M ∇ ( Id − R h ) (cid:0)b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1) , ∇ φ (cid:1) = 0 , ∀ φ ∈ S h . (108) Id denotes the identity application. As in (98) we have Z Ω (cid:16) f (cid:16)b u j +2 ,n +1 h − Γ j b u j,n +1 h (cid:17) − f (cid:0) R h ( b u j +2 ,n +1 − Γ j b u j,n +1 (cid:1)(cid:17) b Θ j +2 ,n +1 h dx ≥ α k b Θ j +2 ,n +1 h k qL q (Ω) − µ k b Θ j +2 ,n +1 h k . Therefore, choosing φ = b Θ j +2 ,n +1 h in (107), we deduce by the triangle inequality, the last inequality and (94)that ( D Θ j +2 ,n +1 / h , b Θ j +2 ,n +1 h ) + γ k∇ b Θ j +2 ,n +1 h k + α k b Θ ,n +1 h k qL q (Ω) ≤ µ k b Θ j +2 ,n +1 h k + (cid:16) Ch r +1 + k (Λ j − Γ j ) D − ( u j,n +1 h − R h u j,n +1 ) k (cid:17) k b Θ j +2 ,n +1 h k (109)4 TITLE WILL BE SET BY THE PUBLISHER
This inequality implies that k Θ j +2 ,n +1 h k − k Θ j +2 ,nh k ≤ kµ k b Θ j +2 ,n +1 h k + k (cid:16) Ch r +1 + (cid:13)(cid:13)(cid:13) (Λ j − Γ j ) D (cid:16) u j,n +1 / h − R h u j,n +1 / (cid:17)(cid:13)(cid:13)(cid:13)(cid:17) , (110)and we deduce, for kµ <
2, that k Θ j +2 ,n +1 h k ≤ k − kµ (cid:16) Ch r +1 + (cid:13)(cid:13)(cid:13) (Λ j − Γ j ) D (cid:16) u j,n +1 / h − R h u j,n +1 / (cid:17)(cid:13)(cid:13)(cid:13)(cid:17) + 2 + kµ − kµ (cid:13)(cid:13)(cid:13) Θ j +2 ,nh (cid:13)(cid:13)(cid:13) . It follows by induction that, k Θ j +2 ,n +1 h k ≤ C (cid:18) kµ − kµ (cid:19) n − j (cid:16) h r +1 + k Θ j +2 ,jh k (cid:17) + k (cid:18) kµ − kµ (cid:19) n − j n X m = j k (Λ j − Γ j ) D ( u j,m +1 / h − R h u j,m +1 / ) k , (111)for n ≥ j , and for 0 ≤ n ≤ j − k ¯Θ j +2 ,n +1 h k ≤ C (cid:18) kµ − kµ (cid:19) n (cid:16) h r +1 + k ¯Θ j +2 , h k (cid:17) + k (cid:18) kµ − kµ (cid:19) n j X m =0 (cid:13)(cid:13)(cid:13) (¯Λ j − ¯Γ j ) D (cid:16) ¯ u j, (2 j +1) m + j +1 / h − R h ¯ u j, (2 j +1) m + j +1 / (cid:17)(cid:13)(cid:13)(cid:13) , (112)where we define ¯Θ j +2 ,nh = u j +2 ,nh − R h u j +2 ,n − ¯Γ j (cid:16) ¯ u j, (2 j +1) n + j +1 h − R h ¯ u j, (2 j +1) n + j +1 (cid:17) . Since n u j,nh o Nn =0 and n ¯ u j,mh o jm =0 are obtained from the same scheme, but for different time steps k and k j = k/ (2 j + 1), respectively, as for (cid:8) u j,n (cid:9) Nn =0 and (cid:8) ¯ u j,m (cid:9) jm =0 , we deduce from the induction hypothesis andthe formulae (17) and (18) that k ¯Θ j +2 , h k = k ¯Γ j (cid:16) ¯ u j,j +1 h − R h ¯ u j,j +1 (cid:17) k ≤ C j X m =0 k ¯ u j,mh − R h ¯ u j,m k ≤ Ch r +1 , (113)and k j X m =0 (cid:13)(cid:13)(cid:13) (¯Λ j − ¯Γ j ) D (cid:16) ¯ u j, (2 j +1) m + j +1 / h − R h ¯ u j, (2 j +1) m + j +1 / (cid:17)(cid:13)(cid:13)(cid:13) ≤ C vuut k j +3 j X m =0 k D (¯ u j,m +1 / h − R h ¯ u j,m +1 / ) k ≤ Ch r +1 . Substituting the last two inequalities in (112), we deduce that k ¯Θ j +2 ,nh k ≤ Ch r +1 , for 0 ≤ n ≤ j, and it follows by the triangle inequality and the induction hypothesis that k u j +2 ,nh − R h u j +2 ,n k ≤ Ch r +1 , for 0 ≤ n ≤ j. (114) ITLE WILL BE SET BY THE PUBLISHER k Θ j +2 ,jh k ≤ Ch r +1 , and we have from (14) and (15) k n X m = j k (Λ j − Γ j ) D ( u j,m +1 / h − R h u j,m +1 / ) k ≤ C √ nk vuut k n + j X m =0 k D ( u j,m +1 / h − R h u j,m +1 / ) k ≤ Ch r +1 . The last two inequalities and (114) substituted in (111) yields k Θ j +2 ,nh k ≤ Ch r +1 , for j ≤ n ≤ N, (115)and it follows from (109) and (114) that k u j +2 ,nh − R h u j +2 ,n k + 2 αk n X i =0 k b u j +2 ,ih − R h b u j +2 ,i k qL q (Ω) ≤ Ch r +2 . (116)Otherwise, proceeding as in the step 1-(ii) of this proof, we choose φ = D Θ j +2 ,n +1 / h in (107) and deduce from(115) that k n X i = j (cid:13)(cid:13)(cid:13) D Θ j +2 ,i +1 / h (cid:13)(cid:13)(cid:13) + γ (cid:13)(cid:13)(cid:13) ∇ Θ j +2 ,n +1 h (cid:13)(cid:13)(cid:13) ≤ Ch r +2 + (cid:16) M ∇ Θ j +2 ,jh , ∇ Θ j +2 ,jh (cid:17) , (117)for j ≤ n ≤ N , and, for 0 ≤ n ≤ j − k j X i =0 (cid:13)(cid:13)(cid:13) D ¯Θ j +2 ,i +1 / h (cid:13)(cid:13)(cid:13) + γ (cid:13)(cid:13)(cid:13) ∇ ¯Θ j +2 ,n +1 h (cid:13)(cid:13)(cid:13) ≤ Ch r +2 (118)since, from Cauchy-Schwartz inequality and (113), we have (cid:12)(cid:12)(cid:12)(cid:16) M ∇ ¯Θ j +2 , h , ∇ ¯Θ j +2 , h (cid:17)(cid:12)(cid:12)(cid:12) ≤ ||| M |||k∇ ¯Θ j +2 , h k ≤ Ch r +2 . By the triangle inequality and the induction hypothesis, inequality (118) for n = j − (cid:12)(cid:12)(cid:12)(cid:16) M ∇ Θ j +2 ,jh , ∇ Θ j +2 ,jh (cid:17)(cid:12)(cid:12)(cid:12) ≤ ||| M |||k∇ Θ j +2 ,jh k ≤ Ch r +2 . Substituting the last identity in (117), we deduce from (118), the induction hypothesis, and the triangle in-equality that k n X i =0 k D ( b u j +2 ,i +1 / h − R h b u j +2 ,i +1 / ) k + γ k∇ (cid:16)b u j +2 ,nh − R h b u j +2 ,n (cid:17) k ≤ Ch r +2 , (119)for 0 ≤ n ≤ N −
1, where C is a constant depending only on j , T , Ω, M , and the derivatives of f and u . Inequality(92) for the case j + 1 follows from (116) and (119). Therefore, we can conclude by induction that the Theoremholds for 1 ≤ j ≤ p + 1. (cid:3) TITLE WILL BE SET BY THE PUBLISHER
Corollary 1.
Under the conditions of Theorem 5, if S h does not satisfy the inverse inequality, provided that,in addition to conditions (2) and (3), f satisfies the inequality | f ( x ) − f ( y ) | ≤ C (cid:0) | x − y | + | x − y | q − (cid:1) , for each x, y ∈ R J , (120) then the solution n u j,nh o Nn =0 , ≤ j ≤ p + 1 , of the scheme (84)-(85) satisfies k u j,nh − u ( t n ) k ≤ C ( h r + k j ) , ∀ n = 0 , , ..., N, k < k . (121) Furthermore, we have the estimate k u j,nh − I rh u j,n k + k n X i =0 k D ( u j,ih − I rh u j,i ) k + 2 αk n X i =0 k u j,ih − I rh u j,i k qL q (Ω) ≤ Ch r (122) where C is a constant depending only on j , T , Ω , M , k , µ , and the derivatives of S , f and u .Proof. Inequality (122) is deduced from Theorem 5 substituting the elliptic operator R h by the interpolatingoperator I rh . By this substitution, the corresponding Claim 1 is obtained from (78) and (79). Since (104) doesnot hold without inverse inequality, (105) is replaced by the inequality (cid:12)(cid:12)(cid:12)(cid:12)Z Ω (cid:16) f ( b u ,n +1 h ) − f ( I rh b u ,n +1 ) (cid:17) (cid:16)b u ,n +1 h − I rh b u ,n +1 (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) k b u ,n +1 h − I rh b u ,n +1 k + k b u ,n +1 h − I rh b u ,n +1 k qL q (Ω) (cid:17) , owing to the hypothesis (120). The order of accuracy in space is reduced since, instead of identities (99) and(108), we have (cid:12)(cid:12)(cid:0) M ∇ ( Id − I rh )( b u j +2 ,n +1 − Γ j b u j,n +1 ) , ∇ φ (cid:1)(cid:12)(cid:12) ≤ C k∇ ( Id − I rh )( b u j +2 ,n +1 − Γ j b u j,n +1 ) kk∇ φ k ≤ Ch r k∇ φ k , for each φ ∈ S h . (cid:3) Numerical experiment
For the numerical experiment we consider the bistable reaction-diffusion equation u t − u xx + 10 u ( u − u − .
25) = 0 in Ω × (0 , T ) ,∂u∂n = 0 on ∂ Ω × (0 , T ) ,u ( · ,
0) = e − x in Ω . (123)We choose Ω = (0 ,
1) and T = 0 . P Lagrange finite elements in space with uniform mesh and the step h = 10 − . We compute areference solution using DC10 with the time step k = 1 . × − (N=1800). Table 3 gives the maximal absoluteerror in time, norm L (Ω) in space, and the order of convergence for each pair of consecutive time steps.For this problem, we have f ( u ) = 10 u ( u − u − . , and inequalities (2) and (3) hold with τ (0) = − µ = 8125 /
3. Therefore, according to Theorem 5,the maximal time step to solve the problem with the DC methods is k = 6 / ≃ . × − , that is N =39 . ≃ t < t < · · · < t N = T , DC N nonlinear systems while DC j , j ≥
2, solves
ITLE WILL BE SET BY THE PUBLISHER j × N systems. For the bistable reaction-diffusion, it is clear that, for N > DC
10 achieves an absoluteerror of about 2 . × − by solving approximately 2250 while DC DC DC DC DC . × − , 2 . × − , 5 . × − and 6 . × − . Sincethe resolution of nonlinear systems is the main burden for these methods, using high order DC methods isadvantageous. Table 3.
Absolute error (order of convergence) for the bistable reaction-diffusion equation N DC2 DC4 DC6 DC8 DC1040 0.115 4.62e-03 9.14e-04 1.97e-04 1.11e-0390 8.48e-04(3.21) 4.59e-05(5.68) 2.05e-06(7.52) 1.55e-06(5.97) 1.45e-06(8.22)180 5.91e-05(3.84) 2.17e-06(7.72) 5.53e-09(8.53) 4.09e-09(8.56) 1.90e-09(9.57)360 3.87e-06(3.93) 8.59e-10(7.98) 2.57e-12(11.07) 4.51e-13(13.15) 8.57e-14(14.44)450 1.55e-06(3.96) 1.44e-10(8.01) 2.33e-13(10.74) 2.40e-14(13.14) 2.48e-15(15.88)900 9.97e-08(4.00) 5.63e-13(7.99) 2.67e-16(9.77) 8.62e-19(14.75) 7.36e-21(18.36)1800 6.25e-09(3.99) 2.18e-15(8.00) 2.13e-19(10.29) 1.74e-22(12.27) –
Acknowledgements
The authors would like to acknowledge the financial support of the Discovery Grant Program of the NaturalSciences and Engineering Research Council of Canada (NSERC) and a scholarship to the first author from theNSERC CREATE program “G´enie par la Simulation”.