A PRIORI AND A POSTERIORI OF A LINEAR ELLIPTIC PROBLEM WITH DYNAMICAL BOUNDARY CONDITION.

Transcription

1 A PRIORI AND A POSTERIORI OF A LINEAR ELLIPTIC PROBLEM WITH DYNAMICAL BOUNDARY CONDITION. Toufic El Arwadi, Séréna Dib, Toni Saya To cite tis version: Toufic El Arwadi, Séréna Dib, Toni Saya. A PRIORI AND A POSTERIORI OF A LIN- EAR ELLIPTIC PROBLEM WITH DYNAMICAL BOUNDARY CONDITION <al v> HAL Id: al ttps://al.arcives-ouvertes.fr/al v Submitted on 11 Jun 015 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 A PRIORI AND A POSTERIORI ERROR ANALYSIS FOR A LINEAR ELLIPTIC PROBLEM WITH DYNAMIC BOUNDARY CONDITION. TOUFIC EL ARWADI, SÉRÉNA DIB, AND TONI SAYAH Abstract. In tis paper, we study te time dependent linear elliptic problem wit dynamic boundary condition. Te problem is discretized by te backward Euler s sceme in time and finite elements in space. In tis work, an optimal a priori error estimate is establised and an optimal a posteriori error wit two types of computable error indicators is proved. Te first one is linked to te time discretization and te second one to te space discretization. Using tese a posteriori errors estimates, an adaptive algoritm for computing te solution is proposed. Finally, numerical experiments are presented to sow te effectiveness of te obtained error estimators and te proposed adaptive algoritm. Keywords. Dynamic boundary condition, finite element metod, a posteriori analysis. 1. Introduction Let IR be a bounded simply-connected open domain in IR, wit a Lipscitz-continuous connected boundary, and let ]0, T [ to denote an interval in IR were T (0, + ) is a fixed final time. We denote by n(x) te unit outward normal vector at x. We intend to work wit te following time dependent linear elliptic problem wit dynamic boundary condition: u(t, x) = 0 in ]0, T [, u (t, x) + β n(x). u(t, x) = 0 on ]0, T [, u(0, x) = u 0 on, were β is a positive constant. Te unknown is u and u 0 is te initial condition at time t = 0. Te solution of problem (1.1) can be represented on te boundary by a Diriclet-to-Neumann semigroup (see for instance [17]). For te existence and uniqueness of tis solution see [17]. In a particular case, were = B(0, 1) te unit ball of R, in is book [14], P.Lax sowed tat te Diriclet-to-Neumann semigroup ad a simple explicit representation. In [9], it is sown tat te Lax representation cannot be generalized if is not te unit ball of R. Tis motivated te autors of [9] and [7] to introduce a semi discrete explicit and implicit Euler s sceme in order to approximate te Diriclet-to-Neumann semigroup numerically. Te convergence of tese semi discrete scemes is based on te Cernoff s product formula. For te discretization of problem (1.1), te autors of [9] sow simple numerical experiments. Te aim of tis work is to sow optimal a priori and a posteriori estimates and some numerical investigations. (1.1) Te idea of te a posteriori error estimates is based on an upper bound of te error between te exact solution and numerical one wit a sum of a local indicators expressed in eac element of te mes. To get more precision and to minimize te error, te goal is to decrease tis indicators by using te adaptive mes tecniques wic consists to refine or coarsen some regions of te mes. Te a posteriori error estimate is optimal if we can make eac one of tese indicators bounded by te local error of te solution June 11, 015. Department of Matematics and computer science, Faculty of Science, Beirut Arab university, P.O. Box: , Beirut,Lebanon, t.elarwadi@bau.edu.lb. Unité de recerce EGFEM, Faculté des Sciences, Université Saint-Josep, B.P Riad El Sol, Beyrout , Liban. s: toni.saya@usj.edu.lb, serena.dib@net.usj.edu.lb. 1

3 T. EL ARWADI, S. DIB, AND T. SAYAH around te corresponding element. We refer for example to te books Verfürt [16] or Ainswort and Oden [1]. For te time dependent problems, we ave two types of computable error indicators, te first one being linked to te time discretization and te second one to te space discretization. We ave to andle te two kinds of indicators, some times, we cange te time step and in an oter times, we adapt te mes. A large amount of work as been made concerning te a posteriori errors. We can cite for example, Ladevèze [1] for constitutive relation error estimators for time-dependent nonlinear FE analysis, Verfürt [15] for te eat equation, Bernardi and Verfürt [6] for te time dependent Stokes equations, Bernardi and Süli [4] for te time and space adaptivity for te second order wave equation, Bergam, Bernardi and Mgazli [5] for some parabolic equations, Ern and Voralïk [10] for estimation based on potential and flux reconstruction for te eat equation and Bernardi and Saya [3] for te time dependent Stokes equations wit mixed boundary conditions,.... In tis paper, te data of te problem is te initial condition of te unknown at te boundary. We propose a very standard low cost discretization relying on te Euler s implicit sceme in time combined wit finite elements in space. Ten, we prove optimal a priori and a posteriori error estimates for te discrete problem. Finally, some numerical simulations are presented based on te proposed algoritm using te FreeFem++ software. Te outline of te paper is as follows: Section is devoted to te study of te continuous problem. In section 3, we introduce te discrete problem and we recall its main properties. In section 4, we study te a priori errors and derive optimal estimates. In section 5, we study te a posteriori errors and derive optimal estimates. In section 6, we sow numerical results of validation.. Analysis of te model In order to write te variational formulation of te problem (1.1), we introduce te Sobolev spaces: H m () = {v L (), α v L (), α m}, equipped wit te following semi-norm and norm : v m, = α =m α v(x) dx 1/ and v m, = v k, k m 1/. As usual, we denote by (, ) te scalar product of L (). For andling time-dependent problems, it is convenient to consider functions defined on a time interval ]a, b[ wit values in a separable functional space, say Y. In te following, f(t) represents te function f(t,.). Let Y denote te norm of Y ; ten for any r, 1 r, we define equipped wit te norm L r (a, b; Y ) = { f measurable in ]a, b[; b ( b 1/r, f Lr (a,b;y )= f(t) r Y dt) a a } f(t) r Y dt <,

4 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 3 wit te usual modifications if r =. It is a Banac space if Y is a Banac space. By te same way, for any integer k, we define { C k (a, b; Y ) = f measurable in ]a, b[ ; } sup t ]a,b[,0 l k f (l) (t,.) Y <. For te existence and te uniqueness of te solution of problem (1.1), we refer to te teorem.1, page 169 in te book [17]. Teorem.1. If is of class C and for eac u 0 L (), problem (1.1) as a unique solution u : [0, + ) H 1 (), satisfying: (1) u C([0, + ); H 1 ()) L ([0, + ); H 1 ()); () u C([0, + ); L ()) C 1 ([0, + ); L ()); (3) n. u C([0, + ); L ()). Furtermore, we ave te following bound: If in addition, u 0 H 1 (), and te unique solution of te problem β u L ([0,+ );L ()) 1 u 0 L (). (.1) u = 0 u = u 0 in on satisfies n. u L (), ten te solution u of te problem (1.1) satisfies (1) u C 1 ([0, + ); H 1 ()); () u C 1 ([0, + ); L ()); (3) n. u C([0, + ); L ()). Remark.. Unfortunately, to our knowledge, tere is no equivalent to te previous teorem in te case of a polyedral domain. Tis will be our next researc work. We suppose tat u 0 H 1/ () and introduce te following variational problem in te sense of distributions on ]0, T [: Find u(t) H 1 () suc tat : u(0) = u 0 on, β u(t, x) v(x) dx + d dt( u(t, s)v(s) ds ) (.) = 0 v H 1 (). Teorem.3. If u L (0, T ; H 1 ()) and u L (0, T ; L ()), te problem (1.1) is equivalent to te variational one (.). Furtermore, we ave te following bound β u L (0,τ,L () ) + 1 u(τ) L () 1 u 0 L (). 3. Te discrete problem From now on, we assume tat is a polyedron. In order to describe te time discretization wit an adaptive coice of local time steps, we introduce a partition of te interval [0, T ] into subintervals [t n 1, t n ], 1 n N, suc tat 0 = t 0 t 1 t N = T. We denote by te lengt of [t n 1, t n ], by τ te N-tuple (τ 1,..., τ N ), by τ te maximum of te, 1 n N, and by σ τ te regularity parameter σ τ = max. n N 1

5 4 T. EL ARWADI, S. DIB, AND T. SAYAH From now on, we work wit a regular family of partitions, i.e. we assume tat σ τ is bounded independently of τ. We introduce an operator π τ by te next definition. Definition 3.1. For any Banac space X and any function g continuous from ]0, T ] into X, π τ g denotes te step function wic is constant and equal to g(t n ) on eac interval ]t n 1, t n ], 1 n N. Similarly, wit any sequence (φ n ) 1 n N in X, we associate te step function π τ φ τ wic is constant and equal to φ n on eac interval ]t n 1, t n ], 1 n N. Now, we describe te space discretization. For eac n, 0 n N, a regular triangulation of (T n ) is a set of non degenerate elements wic satisfies: for eac, is te union of all elements of T n ; te intersection of two distinct elements of T n, is eiter empty, a common vertex, or an entire common edge; te ratio of te diameter of an element κ in T n to te diameter of its inscribed circle is bounded by a constant independent of n and. As usual, denotes te maximal diameter of te elements of all T n, 0 n N, wile for eac n, n denotes te maximal diameter of te elements of T n. For eac κ in T n, we denote by P 1 (κ) te space of restrictions to κ of polynomials wit two variables and total degree at most one. In wat follows, c, c, C, C, c 1,... stand for generic constants wic may vary from line to line but are always independent of and n. For a fixed n N and a given triangulation T n, we define by X n a finite dimensional space of functions suc tat teir restrictions to any element κ of T n belong to a space of polynomials of degree one. In oter words, X n = {v n C 0 (), v n κ is affine κ T n } We note tat for eac n and, X n H 1 (). Tere exists an approximation operator, I L(H (); X n ) suc tat for m = 0, 1 v H (), I (v) v m, C m v,. Te full discrete implicit sceme associated wit te Problem (.) is: Given u n 1 X n 1, find u n wit values in X n solution of v X n, β u n v dx + 1 (u n u n 1 ) v dσ = 0. (3.1) by assuming tat u 0 is an approximation of u(0) in X 0. Remark 3.. It is a simple exercise to prove existence and uniqueness of te solution of problem (3.1) as a consequence of discrete problem of Poisson s equation wit a Robin condition. Teorem 3.3. For eac m = 1,..., N, te solution u m of te problem (3.1) satisfies te bound: u m 0, + u n 1 1, min(1, β) u0 0,, (3.) Proof. For all v X n, let u n be te unique solution of te (3.1). Coosing v (t n ) = u n in (3.1), we find β u n 1, + u n 0, = u n 1 u n dσ. (3.3) By applying te Hölder inequality and summing over n from 1 to m, we get (3.).

6 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION a priori error estimates To get te a priori error estimates, we suppose tat time step and te mes T n don t cange during time iterations. We denote by k te time step, by te parameter of te mes and by X te discrete space. In tis section, te discrete variational formulation (3.1) taken in te time step n + 1, becomes v X, β u n+1 1 v dx + k (un+1 u n ) v dσ = 0. (4.1) To get te a priori error estimate, we need te following te classic Gronwall lemma. Remark 4.1. Gronwall s lemma Let (a n ) n 0, (b n ) n 0 and(c n ) n 0 tree real positive sequences suc tat (c n ) n 0 is an increasing sequence. We suppose tat we ave: (1) () tere exists λ > 0 suc tat: Ten we ave: a 0 + b 0 c 0, (4.) n 1, a n + b n c n + λ n 1 m=0 a m. (4.3) n 0, a n + b n c n e nλ. (4.4) In order to get te a priori error estimate, we begin wit te next teorem. Teorem 4.. ave te bound: If u L (0, T, H ()) and u L (0, T, H ()), and for all m = 0,..., N 1, we I (u(t m+1 )) u m+1 0, + k β I (u(t n+1 )) u n+1 1, (4.5) n=0 C ( + k + u 0 I (u 0 ) 0, ), were C is a constant independent from and k. Proof. We consider te equation (.) for t ]t n, t n+1], take v = v n+1, integrate in time between t n and t n+1, ten take te difference wit (4.1) for v = v n+1 to get tn+1 β (u(t) u n+1 )(x) v n+1 (x) dx dt t n (4.6) + ((u(t n+1 ) u(t n )) (u n+1 u n )) v n+1 )(s) ds = 0. We insert ± (I (u(t n+1 ))) and ± (u(t n+1 )) in te first term, and ±I (u(t n+1 )) and ±I (u(t n )) in te second term, we denote by a n = I (u(t n )) u n and we obtain (a n+1 a n )(s) v n+1 (s)ds + kβ a n 1, = ((I (u(t n+1 )) u(t n+1 )) (I (u(t n )) u(t n )))(s) v n+1 ds tn+1 (4.7) +β (u(t n+1 ) u(t))(x) v n+1 (x) dx dt t n tn+1 +β (I (u(t n+1 )) u(t n+1 )) v n+1 (x) dx dt. t n We denote by T 1 and T respectively te first and second terms of te left and side, and T 3, T 4, T 5 respectively te first, second and tird terms of te rigt and side of te equation (4.7). Ten we coose

7 6 T. EL ARWADI, S. DIB, AND T. SAYAH v n = a n. Te term T 1 can be expressed as Te term T 3 can be bounded as T 3 = = T 1 = We consider te term T 4. We ave T 4 = β β 1 a n+1(s) ds 1 a n(s) ds + 1 (a n+1 a n ) (s) ds. ((I (u(t n+1 )) u(t n+1 )) tn+1 t n tn+1 t n (I (u(t n )) u(t n ))(s) a n+1 (s) ds (I (u(τ)) u(τ)) (s)a n+1 (s) dsdτ I (u (τ)) u (τ) L () a n+1 L ()dτ Ck u L (0,T ;H ()) a n+1 L () C 1 k ε 1 u L (0,T,H ()) + k ε 1 a n+1 0,. tn+1 t n tn+1 tn+1 Finally, te term T 5 can be bounded as t n t (u(t n+1, x) u(t, x))(x) a n+1 (x) dx dt β k u L (0,T,H 1 ()) a n+1 1, u (τ, x) a n+1 (x) dx dτ dt k3 β ε u L (0,T,H 1 ()) + k ε a n+1 1,. T 5 = β tn+1 Using te previous bounds, we get 1 a n+1(s) ds 1 t n (I (u(t n+1 ))(x) u(t n+1, x)) a n+1 (x) dx dt tn+1 βc u(t n+1 ), a n+1 1, dt t n C β k u L (0,T,H ()) k an+1 1, C k β ε 3 u L (0,T,H ()) + k ε 3 a n+1 1,. a n(s) ds + 1 (a n+1 a n ) (s) ds + k β a n+1 1, = C 1 k ε 1 u L (0,T,H ()) + k ε 1 a n+1 0, + k3 β ε u L (0,T,H 1 ()) + k ε a n+1 1, + C k β ε 3 u L (0,T,H ()) + k ε 3 a n+1 1,. (4.8)

8 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 7 We coice ε 1 = 1 8 T, ε = β and ε 3 = β 1 a m+1 0, + k β to get te following bound a n+1 1, n=0 C 3 ( + k ) + 1 a 0 0, + We write te last term of te previous bound as k a n+1 0, = 16 T we suppose tat k 16 T 1 4 n=0 m 1 k 16 T n=0 a n+1 0, + k 16 T a n+1 0,. n=0 k 16 T a n+1 0,, and ten apply te classic Gronwall lemma to get te result. Corollary 4.3. If u L (0, T, H ()) and u L (0, T, H ()), for all m = 0,..., N 1, we ave te following bound: u(t m+1 ) u m+1 0, + k β u(t n+1 ) u n+1 1, (4.10) n=0 C ( + k + u 0 I (u 0 ) 0, ), were C is a constant independent of and k. Proof. For all m = 0,..., N 1: u(t m+1 ) u m+1 0, + k β n=0 u(t n+1 ) u n+1 1, u(t m+1 ) I (u(t m+1 )) 0, + I (u(t m+1 )) u m+1 + k β u(t n+1 ) I (u(t n+1 )) 1, + k β n=0 n=0 I (u(t n+1 )) u n+1 1,. 0, (4.9) (4.11) Based on te teorem 4., te second and of te inequality (4.11) can be bounded by C 1 ( + k ), were C 1 is a constant independent of and k. Te properties of I give te result. 5. a posteriori error estimates We now intend to prove a posteriori error estimates between te exact solution u of Problem (.) and te numerical solution u of Problem (3.1) Construction of te error indicators. In tis section, we will introduce several notations and properties and we will define te indicators. For every element κ in T n, we denote by ε κ te set of edges of κ tat are not contained in, ε m κ te set of edges of κ wic are contained in, κ te union of elements of T n tat intersect κ, e te union of elements of T n tat intersect te edge e, κ te diameter of κ and e te diameter of te edge e, and [ ] e te jump troug e for eac edge e in an ε κ (making its sign precise is not necessary).

9 8 T. EL ARWADI, S. DIB, AND T. SAYAH Also, n κ stands for te unit outward normal vector to κ on κ. For te proofs of te next teorems, we introduce for an element κ of T n, te bubble function ψ κ (resp. ψ e for te edge e) wic is equal to te product of te 3 barycentric coordinates associated wit te vertices of κ. We also consider a lifting operator L e defined on polynomials on e vanising on e into polynomials on te at most two elements κ containing e and vanising on κ \ e, wic is constructed by affine transformation from a fixed operator on te reference element. We recall te next results from [16, Lemma 3.3]. Property 5.1. Property 5.. Denoting by P r (κ) te space of polynomials of degree smaller tan r on κ, we ave { c v 0,κ vψκ 1/ v P r (κ), 0,κ c v 0,κ, v 1,κ c 1 (5.1) κ v 0,κ. Denoting by P r (e) te space of polynomials of degree smaller tan r on e, we ave v P r (e), c v 0,e vψ 1/ e 0,e c v 0,e, and, for all polynomials v in P r (e) vanising on e, if κ is an element wic contains e, L e v 0,κ + e L e v 1,κ c 1/ e v 0,e. We also introduce a Clément type regularization operator C n [8] wic as te following properties, see [, Section IX.3]: For any function w in H 1 (), C n w belongs to te space of continuous affine finite elements and satisfies for any κ in T n and e in ε κ, w C n w L (κ) c κ w 1, κ and w C n w L (e) c 1/ e w 1, e. (5.) For te a posteriori error studies, we consider te piecewise affine function u wic take in te interval [t n 1, t n ] te values u (t) = t t n 1 (u n u n 1 ) + u n 1. Te solutions of Problems (.) and (3.1) verify for t in ]t n 1, t n ] and for all v(t) H 1 () and v (t) X n : (u u ) β (u u )(t, x) v(t, x) dx + (t, s)v(t, s) ds = β (u (t, x) u n (x)) v(t, x) dx β u n u (x) v(t, x) dx (t, s)v(t, s) ds = β t n t (5.3) (u n u n 1 )(x) v(t, x) dx ( u n.n)(x)(v v )(t, x) dx κ T n β κ u n un 1 s(v v )(t, s) ds. We introduce, for every edge e of te mes, te function φ e,n = 1 β [ un.n] e if e ε κ, β u n.n + un un 1 if e ε m κ, (5.4)

10 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 9 Ten, we get te equation β (u u )(t, x) v(t, x) dx (u u ) + (t, s)v(t, s) ds = β t n t (u n u n 1 )(x) v(t, x) dx β φ e,n(x)(v v )(t, x) dx. κ T n e κ e (5.5) Since, we introduce te indicators: For eac κ in T n, (η,κ) = (u n u n 1 ) 0,κ and (ηn,κ) = e φ e,n 0,e. e κ 5.. Upper bounds of te error. We are now able to prove te upper bound. Teorem 5.3. For all m = 1,..., N, we ave te following upper bound β (u u ) L (0,t m,l ()) + u(t m) u m 0, C [ m (ηn,κ) τ + (ηn,κ) + u 0 u 0 ] (5.6) 0,, κ T n κ T n were C is a constant independent of n and. Proof. We denote by L(v) te rigt and side of te equation (5.5) and we introduce te function w(t, x) = e t (u u )(t, x) wic verify te equation w (t, x) + w(t, x) = (u u ) e t (t, x). (5.7) We multiply L(v) by e t and take v = w to obtain e t L(w) = β w(t, x) dx + w (t, s) ds + 1 w (t, s) ds β w(t) 0, + 1 w (t, s) ds. By taking into account tat e t < 1 and remark tat L(w) L(u u ), we ave β w(t) 0, + 1 w (t, s) ds β (u u )(t, x) (u u )(t, x) dx (u u ) + (t, s)(u u )(t, s) ds. (5.8) (5.9)

11 10 T. EL ARWADI, S. DIB, AND T. SAYAH We integrate te last relation in ]t n 1, t n ], sum of n from 1 to m, take into account te relation e t e T to get te following bound tn e [ T β (u u )(t) 0, dt t n u u (t m, s) ds ] tn L(u u ) dt + 1 u u (0, s) ds. t n 1 (5.10) and ten β tm 0 C ( m (u(t) u (t)) 0, dt + 1 u(t m) u m 0, tn were C is a constant independent of n and. (5.11) L(u u ) dt + u 0 u 0 ) 0,, t n 1 Next, we ave to bound te rigt and side of te last inequality. In all te rest of te proof, we denote v = u u and we decompose L(v) = L 1 (v) + L (v) and we bound eac one separately. First, we ave L 1 (v) = β t n t κ T n κ (u n u n 1 )(x) v(t, x) dx β t n t (u n u n 1 ) 0,κ v(t) 0,κ. κ T n (5.1) We integrate te last system in ]t n 1, t n ] and we obtain tn t n 1 L 1 (v) dt κ T n ( β (u n u n 1 ) 0,κ tn t n 1 (t n t) τ n dt ) 1 ( t n v(t) 0,κ dt ) 1 t n 1 β ( (ηn,κ) τ ) 1 ( 3 κ T n C 1 (ε 1 ) (ηn,κ) τ + ε 1 v L (t n 1,t n,l ()). κ T n κ T n v L (t n 1,t n,l (κ)) ) 1 (5.13) Next, we sum over n from 1 to m and get te bound tn t n 1 L 1 (u u ) dt C 1 (ε 1 ) + ε 1 (u u ) L (0,t m,l ()), κ T n (η,κ) (5.14)

12 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 11 were C 1 (ε 1 ) is a constant independent of n and. Next, by taking v (t) = R n, (v(t)), we ave L (v) = β κ T n e κ κ T n e κ C κ T n φ e,n(x)(v v )(t, x) dx e φ e,n 0,e v(t) v (t) 0,e ( e κ e φ e,n ) 1 ( 0,e ( C (ηn,κ) ) 1 ( κ T n κ T n e κ ( C 3 (ηn,κ) ) 1 v(t) 0,, κ T n were C and C 3 are constants independent of n and. We integrate te last system over ]t n 1, t n ] and we ave: tn t n 1 L (v) dt C 3 ( t n t n 1 C 3 ( C 4 (ε ) e κ v(t) 0, e ) 1 v(t) 0, e ) 1 (ηn,κ) ) 1 dt ( t n v(t) 0, dt ) 1 κ T t n 1 n κ T n (η n,κ) ) 1 κ T n (η n,κ) v L (t n 1,t n,l ()) + ε (u u ) L (0,t m,l ()), were C 4 (ε ) is a constant independent of n and. (5.15) (5.16) Te relations (5.11), (5.14) and (5.16) allow us to get te following bound β (u u ) L (0,t m,l ()) + 1 u(t m) u m 0, c [ m (ηn,κ) τ + (ηn,κ) + u 0 u 0 ] 0, κ T n κ T n + (ε 1 + ε ) (u u ) L (0,t m,l ()), were c is a constant independent of n and. By coosing ε 1 = β and ε = β, we get te desired upper bound. (5.17) Next, we will bound te term (u u ) L (0,t m,h 1/ ()). Teorem 5.4. For all m = 1,...N, we ave te bound: (u u ) L (0,t m,h 1/ ()) C [ m [(ηn,κ) τ + (ηn,κ) ] + u 0 u 0 ] (5.18) 0,, κ T n

13 1 T. EL ARWADI, S. DIB, AND T. SAYAH were C is a constant independent of n and. Proof. Let r(t) H 1/ () and consider te problem: { w(t, x) = 0 in ]0, T [, w(t, x) = r(t, x) on ]0, T [. It admits a unique solution w(t) H 1 () wic verify were C 1 is a constant. (5.19) w(t) 0, C 1 r 1/,, (5.0) We consider te equation (5.5), use te relation (5.1) and (5.15), and use te Caucy Scwartz inequality to get 1 (u u ) (t, s)v(t, s) ds v(t) 0, β (u u )(t) 0, + c ( (ηn,κ) ) 1 (5.1) κ T n +β t n t ( ) 1/. (u n u n 1 ) 0,κ κ T n For every v(t) H 1/ (), we consider it lifting in v(t) H 1 () verifying te system (5.19). Using (5.0), we deduce following bound 1 (u u ) (t, s) v(t, s) ds v(t) 1/, 1 (u u ) (t, s) v(t, s) ds v(t) 0, β (u u )(t) 0, + c ( (ηn,κ) ) 1 (5.) κ T n +β t n t ( 1/. (u n u n 1 ) τ 0,κ) n κ T n Ten we get (u u ) 1/, 1 = sup v H 1/ () v(t) 1/, (u u ) (t, s) v(t, s) ds β (u u )(t) 0, + c ( (ηn,κ) ) 1 κ T n +β t n t ( 1/. (u n u n 1 ) τ 0,κ) n κ T n (5.3) We deduce te desired result after integrating over ]t n 1, t n ], summing on n from 1 to m for a m {1,..., N}, and using te teorem 5.3. To conclude te upper bound of our a posteriori error, we bound te term (u π τ u ) L (0,t m,l ()). Teorem 5.5. For all m = 1,...N, we ave te bound (u π τ u ) L (0,t m,l ()) C [ m [(ηn,κ) τ + (ηn,κ) ] + u 0 u 0 ] (5.4) 0,, κ T n

14 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 13 were C is a constant independent of n and. Proof. First, we ave (u π τ u ) L (0,t m,l ()) (u u ) L (0,t m,l ()) + (u π τ u ) L (0,t m,l ()). Te first term of rigt and of te last relation can be bounded, using teorem 5.3, as (u u ) L (0,t m,l ()) C [ m + κ T n (η,κ) κ T n (η n,κ) + u 0 u 0 0, ] 1. (5.5) (5.6) Now, we will bound te second term of te rigt and side of (5.5). For t ]t n 1, t n ], we ave π τ u (t) = u n and ten u (t) π τ u (t) = t t n (u n u n 1 ). (5.7) We obtain te relation (u π τ u )(t) 0, (t t n ) τn ( (u n u n 1 ) (5.8) 0,κ), κ T n tat we integrate over ]t n 1, t n ] and we get tn (u π τ u )(t) 0, 1 (η t n 1 3 n,κ) τ. (5.9) κ T n Finally, we conclude te relation (u π τ u ) L (0,t m,l ()) C [ m + were C is a constant independent of n and. Corollary 5.6. κ T n (η,κ) κ T n (η n,κ) + u 0 u 0 0, For all m = 1,...N, we ave te following upper bound: ] 1, (5.30) (u π τ u ) L (0,t m,l ()) + β (u u ) L (0,t m,l ()) + u(t m ) u m 0, + (u u ) L (0,t m,h 1/ ()) (5.31) C [ m (ηn,κ) τ + (ηn,κ) + u 0 u 0 ] 0,, κ T n κ T n were C is a constant independent of n and. Remark: Estimates (5.31) constitutes our a posteriori error estimate Upper bounds of te indicators. In tis section, we bound te indicators η,κ and η n,κ in order to satisfy te optimality of te a posteriori error. We begin wit te time indicator η,κ. Teorem 5.7. For all m = 1,...N, te following estimate olds ( (ηn,κ) τ C (u π τ u ) L (t n 1,t n,l (κ)) + (u u ) L (t n 1,t n,l (κ)) ), (5.3)

15 14 T. EL ARWADI, S. DIB, AND T. SAYAH were C is a constant independent of n and. Proof. For t ]t n 1, t n ], (5.7) allows us to ave t t n (u n u n 1 )(x) ( (u u )(t, x) + (u π τ u )(t, x) ). (5.33) We integrate te last relation on κ and on ]t n 1, t n ] to get te following result: (η,κ) 6( (u u ) L (t n 1,t n,l (κ)) + (u π τ u ) L (t n 1,t n,l (κ)) ). (5.34) In te following, we will bound te indicators ηn,κ. For t ]t n 1, t n ], We ave β (u(t) u n (u u ) )(x) v(t, x) dx + (t, s)v(t, s) ds = β u n u n u n 1 (t, x) v(t, x) dx (s)v(t, s) ds κ T κ n = β φ e,n(x)v(t, x) dx. Teorem 5.8. were κ T n e κ e For all m = 1,...N, te following bound olds ( (ηn,κ) C (u π τ u ) L (t n 1,t n,l ( κ)) + δ e (u u ) ) (t) L (t n 1,t n,h 1/ (e)), e κ δ e = and C is a constant independent of n and. { 1 if e ε m κ κ 0 elsewere, (5.35) (5.36) Proof. We consider te equation (5.35), an element κ T n and an edge e of κ. We distinguis two cases (1) e ε κ is an interior edge. We set v(t, x) = L e (φ e,n ψ e)(x) in (5.35) and we get (φ e,n) (x)ψ e (x) dx = e (u π τ u )(t, x) L e (φ e,nψ e )(x) dx. e By using te Hölder inequality and te property 5., we get (φ e,n) (x) dx e C (u π τ u )(t) 0, e L e (φ e,n ψ e) 1, e (5.37) (5.38) C (u π τ u )(t) 0, e 1 e φ e,n 0,e, were C, C are constants independent of n and. Ten for all interior edge e we ave e φ e,n 0,e C (u π τ u )(t) 0, e. (5.39)

16 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 15 () e ε m κ is an edge on. We set v(t, x) = L e (φ e,n ψ e)(x) in (5.35) and we get (φ e,n) (x)ψ e (x) dx = e (u π τ u )(t, x) L e (φ e,nψ e )(x) dx κ + 1 (u u ) (t, x)(φ e β,nψ e )(x) dx. By using te Hölder inequality and te property 5., we get e (5.40) φ e,n 0,e C (u π τ u )(t) 0,κ L e (φ e,n ψ e) 1,κ + 1 β (u u ) (t) 1/,e φ e,nψ e 1/,e, (5.41) were C is a constant independent of n and. Te trace teorem and te property 5. allow us to get and ten 1 e φ e,n 0,e C ( (u π τ u )(t) 0,κ + (u u ) (t) 1/,e ), (5.4) e φ e,n 0,e C ( (u π τ u )(t) 0,κ + δ e (u u ) (t) 1/,e ). (5.43) e κ We deduce, by using (5.39) and (5.43), te following bound (ηn,κ) C 1( (u π τ u )(t) 0, κ + δ e (u u ) (t) 1/,e ). (5.44) e κ Finally, by integrating on ]t n 1, t n ], we get (5.36). 6. Numerical results To validate te teoretical results, we perform several numerical simulations using te FreeFem++ software (see [11]). We coose β = 1 and T = a priori error validations. We begin wit te numerical validation of te a priori error estimates. To perform numerical investigations, we need to know te exact solution of problem (.). For tat purpose, we consider instead of a polygon te two-dimensional unit circle wit te following exact solution u(t, x, y) = (e t x) (e t y) + e t y + 1 wic verifies te system (1.1). In fact, te corresponding mes is a polygon and we introduce ere a geometrical approximation. Neverteless, te numerical results given in te end of tis section sow tat tis approximation as not a major influence. (6.1) Figure 1 represents te mes wit m = 50 segments on and a mes step size = π. We coose k = m and we consider te following numerical sceme ( u n+1, v ) + 1 k (un+1, v ) = 1 k (un, v ). (6.)

17 16 T. EL ARWADI, S. DIB, AND T. SAYAH Figure 1. Te mes. We introduce te error err N = N k u n u(t n ) 1, Were N = [ T k ] = [ m ] ([.] is te integer part). π, (6.3) N k u(t n ) 1, Figure sows in logaritmic scale, te error curve between te exact and te numerical solution for different values of te mes step were m takes te values 80, 90, 100, 110, 10. As k =, te error must be of order and te slope of te straigt line must be of order one. Te figure gives a straigt line wit a slope of Error Polyfit E Figure. A priori error curve.

18 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION a posteriori error validations. For te numerical validation of te a posteriori error estimates, we consider te unit square =]0, 1[ and te following initial data on of problem (1.1) { sin(πx) on te top of, u 0 (x, y) = (6.4) 0 on te sides and te bottom of. Te considered numerical sceme is v X n, 1 = u n 1 v (t)dσ. β u n v (t)dx + We introduce te following time and space indicators η = ( and η n = ( κ T n (u n u n 1 κ T n e κ 1 u n v (t)dσ ) ) 1/ 0,κ e φ e,n 1/. 0,e) We begin te iterations wit an initial time step τ 1 = T and an initial mes corresponding to M = 0 0 segments on every side of. Our goal is to validate te a posteriori error estimates. We present ere an adaptive algoritm based on our a posteriori error estimates wic ensures tat te relative energy error between te exact and te approximate solutions is below a prescribed tolerance ε. At te same time, it intends to equilibrate te space and time estimators η n and η. At eac time step, we aim to ave (η ) + (η n) u n 1, (6.5) ε. (6.6) For te adapt mes (refinement and coarsening), we use routines in FreeFem++. We set ε 1 = we introduce te time and space error ε and η e 1 (u n ) = u n 1, η n and e (u n ) = u n. 1, Te actual algoritm is as follows: Coose an initial mes T 0, an initial time step τ 1, and set t 0 = 0 Set n = 1 Loop in time: Wile t n T t n = t n 1 + Solve u n = Sol(un 1,, T n ) calculate ee 1 = e 1 (u n ) and ee = e (u n ) if ((ee 1 > ε 1 ) or (ee ε 1 )) if (ee 1 > ee ) set t n = t n 1 and = / else set t n = t n 1 refine and coarsen te mes using te routine "ReMesIndicator" in FreeFem++, and create new mes called again T n end if else if(ee 1 is very smaller tan ε 1 )

19 18 T. EL ARWADI, S. DIB, AND T. SAYAH set =, u n = un and n = n + 1 set T n = T n 1 else set u n = un and n = n + 1 set T n = T n 1 end if end loop In tis algoritm, if te error does not satisfy te criteria (6.6), te algoritm tests if te time error is larger tan te space error. If so, te algoritm decreases te time step 50%. Oterwise, it adapts te space mes using te indicators and te routine ReMesIndicator in FreeFem++. If te error satisfies te criteria (6.6), te algoritm performs time iterations eiter by increasing te time step if te error is muc smaller tan ε 1, or not keeping te same time step. Figures (3a to 3d) sow te evolution of te mes wit time (ε 1 = 0.01). It is clear tat te mes is concentrated around te part of te boundary were we impose te initial data. (a) Initial mes (b) Mes at t= (c) Mes at t= (d) Mes at t=1 Figure 3. Evolution of te mes during te time iterations. Figures (4a to 4d) sow te evolution of te solution wit time.

20 ERROR STUDIES FOR A TIME DEPENDENT PROBLEM WITH DYNAMIC BOUNDARY CONDITION. 19 (a) Numerical solution for t= (b) Numerical solution for t= (c) Numerical solution for t= (d) Numerical solution for pour t=1 In order to sow te adapt time step, we consider T = 1 and an initial time step τ 1 = Figure 4 sow te evolution of te time step during te time iterations. At t = 0, te algoritm decreases te time step from 0.05 to and during te iterations, te time step increases progressively. Figure 4. Time wit respect to time step.

21 0 T. EL ARWADI, S. DIB, AND T. SAYAH Tese experiments are in very good coerence wit te teoretical results. So tey prove te interest of our approac. References [1] M. Ainswort and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Matematics (New York). Wiley- Interscience [Jon Wiley & Sons], New York, (000). [] C. Bernardi, Y. Maday & F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Matématiques et Applications 45, Springer-Verlag (004). [3] C. Bernardi and T. Saya, A posteriori error analysis of te time dependent Stokes equations wit mixed boundary conditions, IMA Journal of Numerical Analysis, doi: /imanum/drt067, (014). [4] C. Bernardi & E. Süli, Time and space adaptivity for te second order wave equation, Mat. Models and Metods in Applied Sciencesn 15, pp (005). [5] A. Bergam, C. Bernardi and Z. Mgazli, A posteriori analysis of te finite element discretization of some parabolic equations, Mat. Comp. 74, 51, pp (005). [6] C. Bernardi & R. Verfürt, A posteriori error analysis of te fully discretized time-dependent Stokes equations, Mat. Model. and Numer. Anal., 38, pp (004). [7] M.A. Cerif, T. El Arwadi, H. Emamirad and J.M Sac-épée, Diriclet-to-Neumann semigroup acts as a magnifying glass Semigroup Forum 88 (3), pp (014). [8] P. Clément, Approximation by finite element functions using local regularisation, R.A.I.R.O. Anal. Numer., 9, pp (1975). [9] H. Emamirad and M. Sarifitabar, On Explicit Representation and Approximations of Diriclet-to-Neumann Semigroup, Semigroup Forum 86 (1), pp (013). [10] A. Ern and M. Voralík, A posteriori error estimation based on potential and flux reconstruction for te eat equation, SIAM J. Numer. Anal. 48, 1,, pp (010). [11] F. Hect, New development in FreeFem++, Journal of Numerical Matematics, 0, pp (01). [1] P. Ladevèze, Constitutive relation error estimators for time-dependent nonlinear FE analysis, Comput. Metods Appl. Mec. Engrg. 188, 4 (000), IV WCCM, Part II (Buenos Aires, 1998). [13] P. Ladevèze and N. Moës, A new a posteriori error estimation for nonlinear time-dependent finite element analysis, Comput. Metods Appl. Mec. Engrg. 157, 1-, pp (1998). [14] P. D. Lax, Functional Analysis, Wiley Inter-science, New-York, 00. [15] R. Verfürt, A posteriori error estimates for finite element discretizations of te eat equation, Calcolo 40, 3, pp (003). [16] R. Verfürt, A Review of A Posteriori Error Estimation and Adaptive Mes-Refinement Tecniques, Wiley and Teubner Matematics (1996). [17] I. I. Vrabie, C 0 -Semigroups and Applications, Nort-Holland, Amsterdam, 003.