A posteriori error estimates of fully discrete finite element method for Burgers equation in 2-D

Transcription

1 A posteriori error estimates of fully discrete finite element method for Burgers equation in -D Tong Zhang a,b, Jinyun Yuan b, Damazio Pedro b a School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 4543, P.R.China b Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba , P.R.Brazil Abstract Here, a posteriori error estimates of finite element method are derived for Burgers equation in two-dimension. By use of constructing an appropriate Burgers reconstruction, some posteriori estimates in L L, L H and L L norms with optimal order of convergence are established for the semidiscrete scheme. Furthermore, a fully discrete scheme with corresponding posteriori estimates are studied based on backward Euler scheme. Finally, some numerical results are presented to verify the performance of the established posteriori error estimators. Keywords: A posteriori error estimates; Burgers equation; Burgers reconstruction; Backward Euler scheme; Adaptive algorithms MSC: 65N5, 65N3, 76D5.. Introduction We assume that Ω is a bounded polygonal domain in R with a sufficiently smooth boundary Ω for our study in this paper. Consider the following Burgers equations u t ν u + u u = f in Ω, T ], u = on Ω, T ], u = u on Ω {},. where u, f and u are the velocity, the prescribed body force, and the initial velocity respectively with the diffusion coefficient ν >, and the finite time T >. In the last decade years, there is a growing demand of designing and developing reliable and efficient space-time numerical algorithms for the parabolic partial This work was partially supported by CAPES and CNPq, Brazil, and the work of the first author was partially supported by the Doctor Fund of Henan Polytechnic University B-98, China as well. addresses: tzhang@hpu.edu.cn Tong Zhang, yuanjy@gmail.com Jinyun Yuan, pddamazio@gmail.com Damazio Pedro Preprint submitted to??? April 4, 3

2 differential equations. Most of these methods are based on a posteriori error estimators due to the significant advantages of the adaptive method []. Although the theory of a posteriori error estimates of finite element method for elliptic problems is well-developed [3, 8], the theory for parabolic problems is less developed, and only a few results have been published until now, such as the spatial semidiscrete adaptivity [4, 5], the time semidiscrete adaptivity [8, 3] and space-time completely discrete scheme [, 9, 3]. Combining the energy technique with the idea of elliptic reconstruction, Makridakis and Nochetto established an optimal a posteriori error estimates of semidiscrete finite element method for linear parabolic problem in []. Later, based on backward Euler method, Lakkis and Makridakis [] extended these techniques to completely discrete cases. The role of the elliptic reconstruction in a posteriori error estimates is similar to the role played by elliptic projection introduced by Wheeler [33] for obtaining optimal a priori error estimates of finite element method for parabolic problems. Thanks to the elliptic reconstruction ũ, the error u u h u h is the numerical solution can be split into two parts, one is u ũ and the other is ũ u h. The estimates of ũ u h are based on a posteriori analysis of an elliptic problem, while the estimates of u ũ can be controlled by energy arguments in terms of the estimates of ũ u h. This analysis is further developed for linear parabolic problems by maximum norm estimates [], discontinuous Galerkin method [5] or the mixed finite element method []. Although some results about evolution equations have been reported, for nonlinear case, there are still some open research issues. In this paper, we focus on the development of a posteriori error bounds for Burgers equation. Burgers equation is of interest primarily as a model for the unsteady incompressible Navier-Stokes equations. This equation has been investigated by many scientists [3, 6, 3]. There are examples of a posteriori error bounds for the steady Burgers equation [3] and incompressible Navier-Stokes equations [8, 9, 5, 6]. By energy method and an appropriate elliptic reconstruction, Makridakis et. al. have established a posteriori estimates with optimal norms for linear parabolic problem and unsteady Stokes equation in [, 9, ], respectively. Here, based on the techniques developed by Makridakis, we consider a posteriori error estimates of finite element method for two dimensional Burgers equation. By introducing an appropriate Burgers reconstruction, a posteriori error estimates with optimal order of convergence are derived in both semidiscrete and fully discrete formulations. This article is organized as follows. In Section, we formulate the finite element method and recall some basic results. In Section 3, we present a posteriori error estimates of finite element method for spatial semidiscrete formulation. In Section 4, we present the fully discrete formulation and establish the corresponding a posteriori error estimates. Finally, we provide some numerical experiments to verify the performances of established error estimators.

3 . Preliminaries In this section, we present some classical results about Burgers equation and recall some important lemmas which are useful to derive our main results... Basic notations and function setting for Burgers equation For the mathematical setting of problem., we denote X = H Ω, Y = L Ω, DA = H Ω X, where A is the Laplace operator Au = u. Standard notations are used for the Sobolev spaces with the norm and the seminorms in this work see []. For all T > and integer number n, define H n, T ; W s,p Ω = {v W s,p Ω; with the norm given by v H n,t ;W s,p Ω = i n Especially, as n =, we denote the norm as Let with the norm i n [ T T di dt i v s,p,ω dt < }, di dt i v s,p,ω ]. T v,t ;L W s,p Ω = v s,p,ωdt. L, T ; W s,p Ω = {v W s,p Ω; ess sup v s,p,ω < } t T v L,T ;W s,p Ω = ess sup v s,p,ω. t T For the initial data u, we need the following assumption. A. The initial velocity u DA and f, f t L, T ; Y are assumed to satisfy Au + T f + f t dt C. Throughout of whole paper, the letter C > denotes a generic constant, independent of mesh parameter and time step, and maybe different at different occurrences. The continuous bilinear form a, on X X is defined by au, v = ν u, v, 3

4 and the trilinear form by respectively bu, v, w = u v, w u, v, w X. It is easy to verify that b,, satisfies the following important properties for all u, v, w X, where N = sup u,v,w X bu, v, w N u v w,. bu,v,w u v w, and bu, v, w + bv, u, w + bw, u, v C u Av w,. for all u X, v DA, w Y. For the subsequence convenience, we recall the Gronwall lemmas, which will be frequently used in the analysis of a posteriori error estimates see [9, ]. Lemma.. Let gt, ht, yt be three locally integrable nonnegative functions on time interval [,, such that for any fixed time t and all t t yt + Gt C + hsds + t gsysds, t where Gt is a nonnegative function on [,, C is a constant. Then, yt + Gt C + hsds exp gsds. t t With above notations, the variational formulation of problem. reads as: For all t, T ], find u X, such that for all v X u t, v + au, v + bu, u, v = f, v..3 Remark.. For the bounded of the exact solution, it follows from v = u t in.3,. and Lemma. that ν u + u t s ds exp C.. Finite element approximation Au ds ν u + ν f ds. Let h > be a real positive parameter. The finite element subspace X h of X is characterized by T h = T h Ω, a partitioning of Ω into triangles K assumed to be uniformly shape-regular as h, see [] for details. 4

5 Definition.3 Elliptic projection. For v X, we define a projection operator R h X h by av R h, v h =.4 for all v h X h. Furthermore, for any v DA operator R h satisfies see []: v R h + h v R h Ch..5 Now, we present the spatial semidiscrete finite element formulation for problem.: Find u h X h, for all t, T ] such that u ht, v h + au h, v h + bu h, u h, v h = f, v h v h X h..6 Since the bilinear form u h, v h is coercive on X h X h, it generates an invertible operator A h : X h X h through the definition see [6, 7] A h u h, v h = A / h u h, A / h v h = u h, v h, u h, v h X h. By using a similar argument to one used in [3, 4, 7] about the nonlinear convection-diffusion problem, we can obtain the following result. Lemma.4. Assume that Ω is a bounded polygonal domain with a sufficiently smooth boundary Ω, under the assumption of A, there exists a constant C such that T u u h + u u h dt Ch. Furthermore, for the semidiscrete finite element scheme.6, the following smooth properties of u h hold. Lemma.5. Under the assumptions of Lemma.4, there exists a constant C such that Proof. A h u h + ν u h + A h u h s ds + For the L -norm of A h u h, there is A h u h = sup v h L Ω A h u h, v h v h. u ht s ds C. For v h L Ω, we split v h into two parts, i.e., there exist uniqueness vh X h and vh X h, such that v = v h + v h X h Xh = L Ω. Then, according to the definition of A h, one finds A h u h, v h = A h u h, v h + A hu h, v h = A hu h, v h. 5

6 As a consequence, we have A h u h = A h u h, v h A h u h, vh sup = sup v h L Ω v h vh X v h h v h L Ω u u h, vh sup + Au, vh vh X v h h v h L Ω It follows from Lemma.4 that u u h vh sup + Au vh vh X v h h v h L Ω Ch u u h + Au. A h u h + Choosing v h = u ht in.6 and using., we obtain u ht + ν A h u h s ds C..7 d dt u h = f, u ht bu h, u h, u ht f u ht + C A h u h u h u ht f + C A h u h u h + u ht. Integrating with respect to time from to t, using Lemma. and.7, we find that ν u h + u ht s ds exp C C ν u h + ν f ds..8 We end this section by introducing some properties about L -projection operator, which can be found in reference []. Lemma.6. Assume that there exist two L -projection operators I h : X X h and Ih n : X Xn h Xn h will be defined in Section 4, which are defined by ϕ I h ϕ, w h = w h X h or X n h, ϕ X I h takes I h or I n h. Furthermore, if ϕ DA, the following properties hold h i ϕ I h ϕ,k Ch i K ϕ,ω K i =,, ϕ I h ϕ,e Ch / E ϕ,ω K, where ω K = K K K and ω E = E K K for K, K T h. 6

7 3. A posteriori error estimates for semidiscrete formulation In this section, we present a posteriori error estimates of Burgers equation in spatial semidiscrete formulation.6, and establish the computable upper and lower bounds for numerical solution u h in various of norms. Let e u = u h u. It follows from.3 and.6 that e u satisfies e ut, v + ae u, v + bu h, u h, v bu, u, v = rv, 3. where the residual rv is given by rv = u ht, v + ν u h, v + bu h, u h, v f, v. 3. From.6, we know that rv h = for v h X h. We split the error e u into two parts Then, 3. can be rewritten as e u = u h u = ũ u ũ u h = ξ u η u. ξ ut, v + aξ u, v + bξ u, u h, v + bu, ξ u, v = rv + η ut, v + aη u, v + bη u, u h, v + bu, η u, v. 3.3 Now, we introduce the Burgers reconstruction ũ X of u h t for all t, T ]. Definition 3. Burgers reconstruction. For given u h, we define the Burgers reconstruction ũt satisfying aũ u h, v + bη u, u h, v + bu, η u, v = rv. 3.4 Remark 3.. For the given u h, r, from the continuity and coercivity of a,, it is easy to know that 3.4 admits a unique solution ũ X for all t, T ]. By the definition of 3.4, equation 3.3 can be transformed into ξ ut, v + aξ u, v + bξ u, u h, v + bu, ξ u, v = η ut, v Error estimates between exact solution and Burgers reconstruction In this subsection, we derive some estimates about problem 3.5 by using the energy method and some standard techniques in the analysis of parabolic equations. Lemma 3.3. Assume that Ω is a bounded polygonal domain with a sufficiently smooth boundary Ω. Let ξ u be the solution of 3.5. Then, for all t, T ], there exists a constant C such that ν ξ u t + ξ u t + ν ξ ut s ds ξ u s ds C ξ u + C ν / ξ u + η ut s ds, 3.6 η ut s ds

8 Proof. that Choosing v = ξ u in 3.5, by Cauchy inequality and., we obtain d dt ξ u + ν ξ u = η ut, ξ u bξ u, u h, ξ u bu, ξ u, ξ u η ut ξ u + C A h u h + Au ξ u ξ u C A hu h + Au + ξ u + ν η ut + ν ξ u. Kicking the last term, using Lemma.3 and integrating with respect to time from to t, we yield ξ u t + ν ξ u s ds ξ u + η ut s ds + C ξ u s ds. 3.8 The result in 3.6 comes from Lemma.. Secondly, taking v = Aξ u in 3.5 and using Cauchy inequality, one finds d dt ξ u + ν Aξ u = η ut, Aξ u bξ u, u h, Aξ u bu, ξ u, Aξ u η ut Aξ u + C A h u h + Au ξ u Aξ u C ν A hu h + Au ξ u + ν η ut + ν Aξ u. 3.9 Kicking the last term in 3.9, integrating it with respect to time from to t, using Lemmas. and.5. we obtain that ξ u t + ν Aξ u s ds C ξ u + η ut s ds. 3. Thirdly, differentiating 3.5 with respect to time and choosing v = ξ ut, in terms of.,. and Lemma.5, we obtain that d dt ξ ut + ν ξ ut = η utt, ξ ut bξ ut, u h, ξ ut bξ u, u ht, ξ ut bu t, ξ u, ξ ut bu, ξ ut, ξ ut η utt ξ ut + C A h u h + Au ξ ut + Aξ u u ht ξ ut + C Au t ξ u ξ ut C η utt + Aξ u u ht + Au t ξ u + C A h u h + Au + ξ ut + ν ξ ut. Kicking the last term and integrating it with respect to time from to t, using Lemma., 3.6 and 3. we arrive at ξ ut + ν ξ ut ds C ξ u + ξ u + ξ ut + 8 η utt ds + η ut ds. 3.

9 Finally, taking v = ξ ut in 3.5 one finds that ξ ut + ν d dt ξ u = η ut, ξ ut bξ u, u h, ξ ut bu, ξ u, ξ ut η ut ξ ut + C A h u h + Au ξ u ξ ut η ut + C A h u h + Au ξ u + ξ ut. Kicking the last term, using Lemma.5 and integrating with respect to time from to t, one finds ν ξ u t + ξ ut s ds ν ξ u + η ut s ds + C ξ u s ds. By Lemma. and inequality a + b a + b a, b, we obtain the desired result A posteriori error estimates for Burgers reconstruction In this subsection, we derive a posteriori error estimates for Burgers equation in semidiscrete formulation.6. To achieve this aim, we need the following a posteriori estimates about η u and η u related to Burgers reconstruction 3.4. Lemma 3.4. Assume that Ω is a bounded polygonal domain with a sufficiently smooth boundary. Let ũ and u h be the solutions of 3.4 and.6, respectively. Then, there exists a constant C such that /, η u C h 4 K u ht ν u h + u h u h f,k + h 3 E [ν u h ],E 3. K T h E E h /.3.3 η u C h K u ht ν u h + u h u h f,k + h E [ν u h ],E K T h E E h Proof. problem Firstly, consider that Φ DA X is the solution of the elliptic aφ, v + bu, v, Φ + bv, u h, Φ = g, v, 3.4 where u and u h are the solution of. and.6 respectively. According to the Lax-Milgram Theorem, we know that system 3.4 is well-posed and has a unique solution Φ satisfying see [9, ] Φ C g. 3.5 Choosing v = ũ u h in 3.4 and using.4, one arrives at η u, g = ν Φ, ũ ν Φ, u h + bu, ũ u h, Φ + bũ u h, u h, Φ = ν Φ, ũ ν R h, u h + bu, ũ u h, Φ + bũ u h, u h, Φ = ν Φ R h, ũ + bu, ũ u h, Φ R h + bũ u h, u h, Φ R h +ν R h, ũ u h + bu, ũ u h, R h + bũ u h, u h, R h

10 From the definition of 3.4, we know that ν R h, ũ u h + bu, ũ u h, R h + bũ u h, u h, R h = rr h. According to 3. with v = R h X h, equation 3.6 can be rewritten as η u, g = ν Φ R h, ũ + bu, ũ u h, Φ R h + bũ u h, u h, Φ R h. Using.4 again and noting 3.4, we deduce that η u, g = ν Φ R h, ũ + bu, ũ u h, Φ R h + bũ u h, u h, Φ R h = ν Φ R h, ũ u h + bu, ũ u h, Φ R h + bũ u h, u h, Φ R h = rφ R h. 3.7 By.5, 3. and Green s formula, one finds η u, g = u ht ν u h + u h u h f, Φ R h + E E h E [ν u h ] Φ R h ds 3.8 K T h u ht ν u h + u h u h f,k Φ R h,k + E E h [ν u h ],E Φ R h,e C K Th h 4 K u ht ν u h + u h u h f,k + h 3 E [ν u h ],E Φ. E E h Using elliptic regularity 3.5 in 3.8, we deduce that η u, g g C K Th h 4 K u ht ν u h + u h u h f,k + h 3 E [ν u h ],E. 3.9 E E h Taking the supermum over g, we obtain the desired result 3.. By differentiating 3.8 with respect to time, using 3. and 3.4, following the proofs of 3., we can obtain that η ut C h 4 K u htt ν u ht + u ht u h + u h u ht f t,k / K T h + E E h h 3 E [ν u ht ] /, 3. In order to obtain the estimate 3.3, we make the following assumption: ν [ N exp C C + exp C ν Au ds ν u h + ν ] f ds. 3.

11 Taking v = η u in 3.4, and using the fact that r I h η u =, by Green formula and assumption 3., we obtain that ν η u = rη u bη u, u h, η u bu, η u, η u = rη u I h η u bη u, u h, η u bu, η u, η u rη u I h η u + N u h + u η u u ht ν u h + u h u h f, η u I h η u + [ν u h ] η u I h η u ds + N u h + u η u. E E h E By.8 and 3., we finish the rest of proof 3.. As a consequence, combining Lemmas 3.3 and 3.4, we obtain the main theorem of a posteriori error estimates for Burgers equation in semidiscrete formulation. Theorem 3.5. Let Ω be a bounded polygonal domain with a sufficiently smooth boundary, u and u h the solutions of.3 and.6, respectively. Then, for all t, T ] there exists a constant C such that u h u C u h u + η u + η u + u h u C u h u + η u + η u + η ut s ds, η ut s ds, where the estimates of η u, η u and η ut are given by 3., 3.3 and 3., respectively. 4. A posteriori estimates for fully discrete scheme In this section, we consider a posteriori estimates of fully discrete approximation for Burgers equation. based on backward Euler scheme. Set = t < t < < t N = T, I n = t n, t n ] and denote = t n t n. For n [, N], let T n be a refinement of macrotriangulation which is a triangulation of the domain Ω that satisfies the same conformity and shape regularity assumptions made on its refinements see [9] for details. Set h n x = diamk, where K T n and x K. Given two compatible triangulations T n and T n, namely, they are refinements of the same macrotriangulation, let ˆT n be the finest common coarsening of T n and T n, whose mesh size is given by ĥn = maxh n, h n. For more information, we can refer to Appendix A of reference []. Denote t ϕ n := ϕ n ϕ n, f n = ft n.

12 We set X n h as the finite element subspace of X defined over the triangulations T n. Given U = I n h u, find {U n } with U n X n h at t = t n. For v X n h as n = and for n [ : N] ν U, v + bu, U, v = f, v, 4. U n U n, v + ν U n, v + bu n, U n, v = f n, v. 4. For the existence, uniqueness and convergence of U n of problem 4., following the guidelines provided in [3, 4], we have the following results. Theorem 4.. Assume that Ω is a bounded polygonal domain with Lipschitz continuous boundary, under the assumption of A. Then, there exists a constant C such that T ν u U n + ν u U n dt Ch, A h U n + ν U n + A h U n s ds C. Using a sequence of discrete values {U n }, n =,,,..., N, we define a continuous piecewise linear function Ut for t [, T ] by Ut = t t n U n + t t n U n, t n < t t n, n =,,..., N. 4.3 Note that the time derivative of U restricted to I n is U t In = t U n for t I n. 4.4 To motivate the use of Burgers reconstruction, we denote e u t = Ut ut. For v X n h, e u satisfies e ut, v + ν e u, v + bu, U, v bu, u, v = ν U U n, v + f n f, v + Ih n k U n U n, v + U n Ih n n k U n, v n +ν U n, v f n, v + bu, U, v + bu n, U n, v bu n, U n, v. 4.5 Define the residual r n for n =,,..., N as r n v = U n I n h U n, v + ν U n, v + bu n, U n, v f n, v. 4.6 Definition 4. Burgers reconstruction. For given U n n =,,..., N, find ũ n X for v X such that ν ũ n U n, v + bũ n U n, U, v + bu, ũ n U n, v = r n v. 4.7

13 Combining 4.6 with 4. and Lemma.6, we know that r n v h = for v h X n h. Note that U n is Burgers reconstruction of ũ n at t = t n. Using a sequence of discrete values {ũ n } n =,,..., N, for t [, T ], we define a continuous function of time as the continuous piecewise linear interpolation ũt: ũt = t t n ũ n + t t n ũ n, t n < t t n, n =,,..., N. 4.8 Furthermore, for v X and t [, T ], ũ satisfies ν ũ U, v + bũ U, U, v + bu, ũ U, v = rv, 4.9 where rv is piecewise linear interpolation of {r n } N n=. In order to use 4.7, we split the error of e u into two parts e u = ũ u ũ U ξ u η u. 4. Thanks to 4.7, equations 4.5 can be rewritten as for v X Note that ξ ut, v + ν ξ u, v + bξ u, U, v + bu, ξ u, v = η ut, v + ν ũ ũ n, v + f n f, v + I n h U n U n, v +bu n u, U U n, v + bũ ũ n, U, v + bu, ũ ũ n, v. 4. ũ ũ n = t n t ũ n ũ n = t n t t ũ n, 4. U U n = t n t U n U n = t n t t U n. 4.3 From equation 4.7, we deduce that As a consequence, one finds ν ũ n, v + bũ n, U, v + bu, ũ n, v = ν U n, v r n v + bu n, U, v + bu, U n, v. 4.4 ν ũ n ũ n, v + bũ n ũ n, U, v + bu, ũ n ũ n, v 4.5 = ν U n U n, v r n v r n v + bu n U n, U, v + bu, U n U n, v. Substituting 4.3 and 4.5 into 4., we obtain that ξ ut, v + ν ξ u, v + bξ u, U, v + bu, ξ u, v = η ut, v + Ih n k U n U n, v + f n f, v + t n t r n v r n v n ν U n U n, v bu n U n, U, v bu n, U n U n, v.4.6 3

14 Next, we present some estimates about the errors between Burgers reconstruction 4.7 and the exact solution of.3 in various norms. In order to simplify the expressions, we introduce some notations as follows. Let E m = E m 3 = E m 4 = n= n= n= n t n f n f ds, E m = k n h n I I n h U n, k 3 n n= n t n η ut s ds, ĥn t r n + ν t U n + A h U + A h U n t U n, E m 5 = h I h I k U + n= k n ĥn t I n h I U n + h m I m h I k m U m. Theorem 4.3. Assume that Ω is a convex polygonal domain. Let u and U be the solutions of.3 and 4., respectively. Then, for m [, N] there exists a constant C such that m [ ξ u t m + ν ξ u s ds C e u + η u + j [ ν ξ u + ξ ut ds C e u + η u + 4 i= 5 i= ] Ei m, 4.7 ] Ei m. 4.8 Proof. Choosing v = ξ u in equation 4.6, and using Lemma.6, obtain that d dt ξ u + ν ξ u = η ut, ξ u + f n f, ξ u + Ih n k U n U n, ξ u bξ u, U, ξ u bu, ξ u, ξ u n +t n t t r n ξ u ν t U n, ξ u b t U n, U, ξ u bu n, t U n, ξ u = η ut, ξ u + f n f, ξ n + Ih n k U n U n, ξ u Ih n ξ u bξ u, U, ξ u bu, ξ u, ξ u n +t n t t r n ξ u ν t U n, ξ u b t U n, U, ξ u bu n, t U n, ξ u T n + T n + T n 3 + T n 4 + T n 5 + T n Now, we estimate the right-hand side terms of 4.9 separately. For T n and T n, with Cauchy inequality, it is easy to see that η ut, ξ u + f n f, ξ n η ut ξ u + f n f ξ u η ut + f n f + ξ u. 4

15 For T3 n, by Cauchy inequality and Lemma.6, we find that For T n 4 T3 n kn Ih n U n U n ξ u Ih n ξ u 3 ν k n h n I Ih n U n + ν 6 ξ u. and T 5 n, by. and Theorem 4., we arrive at T n 4 + T n 5 C A h U + Au ξ u ξ u 3C ν A hu + Au ξ u + ν 6 ξ u. For T6 n, using the fact that rn v h = for v h X h, we have r n rn v h = for all v h X n X n. Let În h be the L -projection relative to the finest common coarsening ˆT n of T n and T n. For t t n, t n ], we deduce that [ ] T6 n t n t t r n ξ u În h ξ u ν t U n, ξ u b t U n, U, ξ u bu n, t U n, ξ u 3 ν C A h U + A h U n t U n + ĥn t r n + ν t U n Combining above inequalities with 4.9, integrating it with respect to time from to t m with m [ : N], and using Lemma., we complete the proof of 4.7. Next, we choose v = ξ ut in 4.6 and obtain ξ ut + ν d dt ξ u + bξ u, U, ξ ut + bu, ξ u, ξ ut = η ut, ξ ut + f n f, ξ ut + Ih n k U n U n, ξ ut + t n t t r n ξ ut n ν t U n, ξ ut b t U n, U, ξ ut bu n, t U n, ξ ut = η ut, ξ ut + f n f, ξ ut + Ih n k U n U n, ξ ut Ih n ξ ut n +t n t t r n ξ ut t U n, ξ ut b t U n, U, ξ ut bu n, t U n, ξ ut T n + T n + T n 3 + T n Integrating 4. with respect to time from to t m with m [ : N], one gets Set m ν ξ u t m + = ν ξ u + F t i = n= n m ξ ut ds + bξ u, U, ξ ut + bu, ξ u, ξ ut ds t n T n + T n + T n 3 + T n 4 ds ν 6 ξ u. i max ν ξ u t m + ξ ut t i t m ds, for i [ : m]. 4. 5

16 By Cauchy inequality and integration by parts, it is straight forward to deduce that T n + T n + T4 n [ η ut + f n f + t n t t r n + t U n +C A h U + A h U n t U n ] ξ ut. By using Cauchy inequality again, we get n= n {[ m m + t n T n + T n + T n 4 ds kn 3 n= n n= C A h U + A h U n t U n + t r n + t U n / m η ut ds + t n n= n / } f n f ds t n ] / m /. ξ ut ds For T n, using integration for ξ ut from t n to t n, and then applying summation by parts, we arrive at = = n= n= n t n I n h U n U n, ξ ut ds I n h U n U n, ξ n u ξ n u I m h I + n= Ih I U, ξu k U m, ξu m k m t Ih n I U n, ξu n. From., we derive that m bξ u, U, ξ ut + bu, ξ u, ξ ut ds m C A h U + Au ξ u ξ ut ds m m C A h U + Au ξ u ds ξ ut ds /. 6

17 By Lemma.6 and 4., one deduces that = n= n t n I n h U n U n, ξ ut ds I m h I + C + C n= k m U m, ξ m u I m h ξm u t Ih n I U n, ξu n Ih I U, ξu Ih k ξ u În h ξn u h I h I k U ξ u + h m I m h I n= ĥn t Ih n I U n ξu n h I h I k U + n= + h m I m h I k m U m F t i. k m U m ξ m u ĥn t I n h I U n Combining above inequalities with 4., Theorem 4. and replacing t m by t i, we complete the proof of 4.8. Since 4.9 is quite similar in form to 3.4, we can prove the error estimates similar to Lemma 3.4 by following closely to the proof of Lemma 3.4. Here, we omit the proof. Using required estimates of η u and η ut in Theorem 4., we obtain the final theorem of this section. Now, we firstly introduce some notations to be used in what follows. Set E m 6 = E E h h 3 E [ U ],E + K T h h 4 K r,k, E m 7 = E E h h 3 E [ U m ],E + K T h h 4 K r m,k, E m 8 = E m 9 n= E E h h 3 E t [ U m ],E + = E E h h E [ U m ],E + K T h h K r m,k. K T h h 4 K t r m,k, Theorem 4.4. Under the assumptions of Theorem 4.3 and A, set u and U be the solution of.3 and 4., respectively. Then, for m [, N], the following 7

18 estimates hold [ U m ut m C e u + E m + E3 m + E4 m ] + 8 i=6 m [ U u + U t u t ds C e u + E m + 5. Numerical experiment E m i, 6 i=3 E m i + E m 8 + E m 9 In this section, we provide some numerical results to verify the performance of the established posteriori error estimators. For the computed quantities like errors and indicators, we denote e u = u u h, the error indicator η and the number of triangulares in T h NT which are output of the adaptive algorithm for a given tolerance ε. The experimental convergence rates are given by α eu = log[ e uε / e u ε ], α η = log[ηε /ηε ] log[nt ε /NT ε ] log[nt ε /NT ε ]. The effectiveness index is defined as ration of a posteriori error bound and an approximate norm of the actual error, i.e., e u /η. For a good estimator, this quantity should be a constant, independent of the mesh sizes and the time steps. Although our theoretical findings do not include a proof of efficiency, numerical experiments provide evidences of the efficiency of the estimators. For the space-time algorithm, we follow the guideline provided in []. In order to keep the completeness of our work, we present the outline as follows. Algorithm. Let T be a regular triangulation and ε the given tolerance. i. Compute on the shape-regular partition T with t =. ii. Set an initial time step k, using U to compute u Uh and η on T. iii. Begin the time loop with obtained T n, T n and U n Set the time step t n = mint n +, T, Use to compute Û n on T n, and then compute η n, 3 Set t n = mint n + t n t n ε/ η n, T, and obtain that = t n t n, 4 Use T n, U n and to compute U n and η n, 5 Adapt mesh T n to obtain T n. Use U n and to compute U n, 6 For the next iteration, denote T n+ =T n and + =. iv End the time loop and finish the computation. ], Remark 5.. For simplifying the computation, we adopt Oseen iteration to treat the nonlinear term, and set all the constants C involved in indicator and ν equal to. For the time steps, as the strategy used in [4], we choose a maximum over the time step instead of summing up over different time steps. 8

19 Consider the nonlinear parabolic problem u t u + u u = f in Ω, ] u = g on Ω, ] u = on Ω {}, with Ω = [, ] and T =. Choose f and g with the exact solution ux, y = { sinπt exp x, y 3 cos π }, sinπt 4 t where x, y = x + y /. We adopt the linear polynomial to seek the exact solution by backward Euler scheme in time. Table presents the errors and convergence order with different tolerances ε. As expected, we can see that as ε becomes small, the errors go down while the numbers of triangulares increase quickly. The effectiveness index approaches.6, which is a constant independent of N T and time steps. Next, fixed the tolerance ε =.4, the meshes, the profiles of u h and numerical solutions u h are described in Figures -4 at different times. From these Figures, we can see that more meshes are concentrated on the critical region where the solution vary sharply. Figure 5 expresses the variation of time step as time increases with different tolerances ε. Generally speaking, the smaller of ε, the smaller of time step. Furthermore, as the time increases, the time step becomes smaller and smaller. Finally, we compare the profiles of solutions, including the exact solution and numerical solution with different ε in Figure 6. From these figures we can see that as the tolerance ε decreases, we can simulate the exact solution more precisely. Table : Results obtained using space-time algorithm with linear polynomial. ε NT e u η e u /η α eu α η References [] R.Adams, Sobolev Spaces, Academic Press, New York, 975. [] M.Ainsworth, J.Oden. A Posteriori Error Estimation in Finite Element Analysis. A Wiley-Interscience Publications,. [3] I.Babuska, T.Strouboulis, The Finite Element Method and Its Reliability, Clarendo Press, Oxford,. [4] I.Babuska, S.Ohnimus, A posteriori error estimation for the semidiscrete finite element method of parabolic partial differential equations, Comput. Methods Appl. mech. Engrg,

20 IsoValue IsoValue e a b c Figure : Meshes, the profiles of u h and numerical solution u h at time t= a b c Figure : Meshes, the profiles of u h and numerical solution u h at time t= [5] I.Babuska, M.Feistauer, P.Solin, On one approach to a posteriori error estimates for evolution problems solved by the method of lines, Numer. Math., [6] W.Bangerth, R.Rannacher, Adaptive Finite Element Methods for Diffrential Equations. Birkhauser Verlag, 3. [7] C.Bernardi, T.Sayah, A posteriori error analysis of the time dependent Stokes equations with mixed boundary conditions, Preprint, hal ,v-3 Dec. [8] C.Bernardi, R.Verfürth, A posteriori error analysis of the fully discretized time dependent Stokes equations, ESAIM: Math. Model. Numer. Anal., [9] S.Brenner, L.Scott, The Mathematical Theory of Finite Element Methods, Spring-Verlag, 994. [] P.Ciarlet, The finite element method for elliptic problems, Amsterdam: North- Holland, 978.

21 IsoValue IsoValue a b c Figure 3: Meshes, the profiles of u h and numerical solution u h at time t= a b c Figure 4: Meshes, the profiles of u h and numerical solution u h at time t=. [] A.Demlow, O.Lakkis, C.Makridakis, A posteriori error estimates in the maximum norm for parabolic problem, SIAM J. Numer. Anal., [] K.Eriksson, C.Johnson, Adaptive finite element methods for parabolic problems, I: A linear model problem, SIAM J. Numer. Anal., [3] J.Frutos, B.Garcia-Archilla, J.Novo Nonlinear convection-diffusion problems: fully discrete approximations and a posteiroir error estimates, IMA J. Numer. Anal., [4] B.Garcia-Archilla, E.S.Titi, Postprocessing the Galerkin method: the finite element case, SIAM J.Numer. Anal., [5] E.Georgoulis, O.Lakkis, A posteriori error control for discontinuous Galerkin method for parabolic problems, Preprint arxiv:84.46v [math.na] 8. [6] J.Heywood, R.Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, part I: regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., [7] J.Heywood, R.Rannacher, Finite element approximation of the nonstationary

22 IsoValue IsoValue IsoValue t t a t t b Figure 5: The variable of time step with different ε. a ε =.6, b ε =.4. a b c Figure 6: The profiles of exact solution and numerical solution. a Exact solution, b ε =.4, c ε =.8. Navier-Stokes problem, part II: Stability of solutios and error estimates uniform in time, SIAM J. Numer. Anal., [8] C.Johnson, Y.Y.Nie, V.Thomee, An a posteriori error estimate and adaptive time step control for a backward Euler discretization of a parabolic problems, SIAM J. Numer. Anal., [9] F.Karakatsani, C.Makridakis, A posteriori estimates for approximations of time-dependent Stokes equations, IMA J. Numer. Anal., [] O.Lakkis, C.Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp., [] C.Makridakis, R.H.Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal.,

23 [] S.Memon, N.Nataraj, A.K.Pani, An a posteriori error analysis of a mixed finite element Galerkin approximation to second order linear parabolic problems, SIAM J. Numer. Anal., [3] R.H.Nochetto, G.Savare, C.Verdi, A posteriori error estimates for variable time step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., [4] R.H.Nochetto, G.Savare, C.Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp., [5] S.Nicaise, N.Soualem, A posteriori error estimates for a nonconforming finite element discretization of the time-dependent Stokes problem, J. Numer. Math., [6] S.Nicaise, N.Soualem, A posteriori error estimates for a nonconforming finite element discretization of the time-dependent Stokes problem, II: analysis of the spatial estimator, J. Numer. Math., [7] V.Thomée, Galerkin finite element method s for parabolic problems, North- Holland, Spring-Verlag, 6. [8] R.Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh- Refinement Techniques. Wiley Teubner, 996 [9] R.Verfürth, A posteriori error estimates for nonlinear problem: L r, T ; W,ρ Ω error estimates for finite element discretization of parabolic equations, Numer. Methods PDEs, [3] R.Verfürth, A posteriori error estimates for nonlinear problem: L r, T ; L ρ Ω- error estimates for finite element discretization of parabolic equations, Math. Compt., [3] R.Verfürth, A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations, Calcolo, [3] K.Veroy, C.Prud homme, A.T.Patera, Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds, C.R. Acad. Sci. Pairs, Series I, [33] M.F.Wheeler, A priori L error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal.,