 Applied Mathematics, 2011, 2, 981-986 doi:10.4236/am.2011.28135 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM A Posteriori Error Estimate for Streamline Diffusion Method in Solving a Hyperbolic Equation Davood Rostamy, Fatemeh Zabihi Department of Mat hematics, Imam Khomeini International University, Qazvin, Iran E-mail: rostamy@khayam.ut.ac.ir Received December 23, 2010; revised May 30, 2011; accepted June 7, 2011 Abstract In this article, we use streamline diffusion method for the linear second order hyperbolic initial-boundary value problem. More specifically, we prove a posteriori error estimates for this method for the linear wave equation. We observe that this error estimates make finite element method increasingly powerful rather than other methods. Keywords: Streamline Diffusion Method, Hyperbolic Problems, Wave Equations, Finite Element, A Posteriori Error Estimate 1. Introduction The wave equation based on rigorous a posteriori error estimates is a largely subject, despite the importance of these problems in the modeling of a number of physical phenomena. A posteriori have made every method in-creasingly powerful; such that there are various ap-proaches to a posteriori error estimates and it has new successfully applied to varied problems by several au-thors (see Ainsworth and Tinsley Oden ; Asadzadeh ; Gergouli ; Johnson  and ). Gergouli et al.  and his teammates applied finite element method for linear wave equation and obtained a posteriori error estimates in L (L2) norm in Johnson proved existence solution for second order hyperbolic problems and used discontinuous Galerkin method for them and obtained a priori and a posteriori error esti-mates. In this paper, we do new work and use streamline diffusion method (SD-method) for solving the linear second order hyperbolic initial-boundary value problem. Streamline diffusion methods (Asadzadeh ; Asadza- deh and Kowalczyk ; Eriksson and Johnson ; Brenner ; Dubois ; Fuhrer  ) perform slightly better than the standard finite element methods for smooth solutions and non-smooth solutions hyperbolic problems as a two-dimensional one which both is higher order accurate and has good stability properties. Due to the fact that artificial diffusion is added only in the char-acteristic direction so that internal layers are not smeared out, while the added diffusion removes oscillations near boundary layers. We consider the linear second order hyperbolic initial boundary value problem (see Codina ; Haws, ; Gergoulus et al., ; Iraniparast ; Kalmenov ; in Sobolov space Adams ; Shermenew ) as follows:  01in 0,,0on 0,0,0on 0,00on(0,]tttuauf Tuxu xux uxux T  (1) Here, d is a bounded open polygonal domain with boundary  and we have 1200 1,uH uL, a is a scalar-value function in C and 220, ;fL TL. For (1), we use one variable changing and obtain a new problem. We apply SD-method for new problem and obtain a posteriori error estimates. A posteriori error bound provides a computable upper bound on the error in some norm using the computed finite element solution (see Ainsworth and Tinsley Oden ; Asadzadeh ; Burman ; Johnson and Szepessy ; Sandboge ). In order to make use of the theory of Semigroups we write the system (1) in the following abstract form: 982 D. ROSTAMY ET AL.   2202in 0,,0on00,0on(0, ]twAwFTwxw xwt T  (2) Here, we assume for and tvudx[0, ]tT, also:  T,,,,wxtuxt vxt,  T,,,,tttwxtuxtvxt  T0010,0wx uxuxaIA and  T,0,,Fxtfxt where, I is identity matrix. The rest of this study is organized as follows. In Sec-tion 2, we define slabs for space-time domain and obtain SD-method for (2) this slabs. In Section 3, we consider a posteriori error estimates for SD-method form of Section 2 and obtain dual problem. In Section 4, we define in-terpolation estimates for dual problem. In Section 5, we complete proof for a posteriori error estimates by using definitions in Section 4. 2. The Streamline Diffusion Method In this section, we consider the SD-method for solving (2) that is based on using finite element over the space-time domain . To define this method, let 01 be a subdivision of the time in-terval into intervals [0,]TNtt[0,]T0t T 1,nnnItt, 1N, with time steps 1nn n, and introduce the corresponding space-time slabs, i.e.: kt t 0,1,n1,: ,nnSxtxtttn (3) for . Further, for each n let be a finite element subspace of 0,1, ,1nNnWn 11nHSHS, (see Ad-ams, ) and let: |0, 0 , for nn nWwWwt tI  (4) We can formulate SD-method on the slab for (2), as follows: nSFor , find such that: 0, ,1nN nwW,,,nn nttnntnnwAwg gAgwgFggAgw g ,n (5) where, Chwith C is a suitable chosen (suffi-ciently small, see Johnson, ) positive constant and parameter h is defined in the following. Further, we de-fine the following notations for (6): ,dTnSnwgwgxtd,dn ,,.Tnnwgwxtgxtx 00, lim,limsswxt wxts 0,lim ,swxt wxts The terms including , in the above formula is a jump conditions which imposes a weakly enforced con-tinuity condition across the slab interfaces, at tn and is the mechanism by which information is propagated from one slab to another. For more concisely, after summing over n, we may rewrite (5) as follow: We assume and find|, such that: 10NnnwWww,Bwg Lg (6) For gw and where the bilinear form ., .B and the linear form .L1N define by: 0101,,[],nnttnnNnnnnBw gwAwggAgwg wg where, we define such that for T12,www1, 2i:  ,, 12,,iiiwwww ww T and  1000,,NnnLgFggtAgw g  For ,we define such that be a triangulation of the slab n into triangles K satisfying as usual the minimum angle condition (Ciarlet ) and assume that the parameter h is represented with the maximum di-ameter of the triangles 0hSnhTnhKT. We introduce:   11:for,0,0fornhnnkknhnWwHSHSwPK PKTw ttI k where, kPK denotes the set of polynomials in K of degree less than or equal k and: 10NnhhnWW Thus (6) can be formulated as follows: Find hwWh such that: ,hBw gLg (7) for hgW. Moreover, we know that the exact solution of (6) satisfies: Copyright © 2011 SciRes. AM D. ROSTAMY ET AL. 983fo,BwgLg rgwand by use (6) and (7), we have the Galerkin ogonorthality relation: ,0Beg (8) where, . An a Posteriori Error Estimate for the this section, we shall consider the following simplified (9) where, heww . 3SD-Method Inversion of SD-method for (7) with  = 0: Find whWh, such that for 0,1, ,1nN : 1N1,,011000,[], ,,,Nnn n nht hhnhnnnNnnwAwgwg wgFgw g 0hgW and mplicittake 0,0hw. For siy, we 00w and . In order to the error, weine (10) and denotes the adjoint of the operator L defined in nd 0F con obtain a representation of sider the following auxiliary problem, referred to as the linearized dual problem: Find  such that:  1*0,0, 0,,0,TtLAttTxT x  *L (2) a is a positive weight function. Note that this problem is computed “backward”, but there is a corre-sponding change in sign. Further, we shall first introduce the following notation: 21/2() ,Lwww (11) Multiplying (10) by e and integrating by parts and summing over n, we obtain the following error represen-tation formula: 121()101100,,*,,,LTNtnnNNTtnnnneeeeLeAeeA    (12) We have for by part integrating: d(13) We define and 0,1, ,1nN  1010,dd,,d d,,d,nnTttSnNTTnn tSnNTnntnneextextxt xextextxtx e   T12,eeeT12, andta ob-in for 0,1nN, ,1 :  122121122121 2121 21,dd0()dd10.() dd(())dddd()dd ,nnnnnnnTSnTSSSSSTnSeAeAxtaexaee xteae xtaeextaeextAex tAe    TTddt (14) We define: According to (9), 10102112101100,,,,,,,,,,,,[],,[ ]TnnnNNNNNNNnnnnJextxtdxeeeeeeeeeeee               N., 0NtT and since 0e00w, we get: 1N0[],hnnJw (15) Then in (12), by use (15) and (16), we have: 1211 12NN N () 00 0,,[],thnnLnn neeAew    So that recalling (9) and using the Galerkin orthogo-nality (8), we obtain: Copyright © 2011 SciRes. AM 984 D. ROSTAMY ET AL. 12,Nee121 1() 00 01011,0010,[],,[], ,[],N NthnnLnn nNhhtnnNNhn hthnnnNhnnAewww AwwwFwAww       (16) where, is an interpolant of The idea isate utes for the Dual e consider our interpolant in (16) and ˆhW . now to estim in terms of −1esing a strong sta-ˆes for bility estimatsolution  of the dual problem. . Interpolation Es tim a 4Solution shall nowWˆhW, nameto be the space-time L2-projection of ly if the first, we define the L2-projections: 2:()PL Wnnh  20,2::is constant on , nn nnnLSwLS wxIx ,.in space and in space time, respectively, by: n,d d,TTnhPwxwx w  W 0,1|.,d,nnn InwSw t twk nThen, we can defineˆnnhSW by letting: ˆnnnnhSP W  n nPwhere, nS . Further, if we introduce P and  de-: fined bynn nPSP S and nn nSS respectively, then we can let to Now, we define residual of computed solutioby: nˆhW be: ˆhPPW  nhw0h,th RFwAw  1, ,,onnnhhRw wS ,2,onnnhnnPIwRSk where, I is the identity operator. In the end of this section, we shall give a lemma for projection operators P, leinterpolation estimates by the aving the overall of I and II to next section. Lemma 1: There is a constant C such that for residual RL2: 1222() (),xxLLRP ChIPR   (17) Proof: (see Johnson and Szepessy  and Sa). Completion of the Proof of a n osteriori error esti-ate by estimating of the terms I and II in the error rep-ndboge 5. ThePosteriori Error Estimates this section we state and prove a pImresentation formula (16). To this approach we introduce the stability factors (see Burman ) associated with discretization in time and space, defined by: 2()tLt 12()eLe  (18) and 212()()xx LxeLe (19) respectively. We now apply the result of the previous sections; using Catchy-Schwartz inequality in (16) cou-pled with the interpolation estimate (17) and the strong stability factors (18) and (19), to derive the 22LL a posteriori error estimates for the scheme (9). Theorem 1: The error heww , where w so-lution of the continuous problems (2) and wis the that of (9), sahtisfies the following stabte: ility estima1122220() ()xeLLeChIPR C  111221()222() ()xenLxxeenLLkRhR kR  (20) Proof: Using the notation introduce above, we write (17) as: may 1112()[]NNLwe2000ˆˆ,,()hnnnnnnRkkIII  Below we shall estimate the terms I and II separately. Copyright © 2011 SciRes. AM D. ROSTAMY ET AL. 985Splitting the interpolation error by writing and , we have: ˆˆPP  ˆnnP 12210011()ˆnnNNIRPP000020() ()ˆ,,NnnxxLLRP RPCh I PR   where we have used the fact that is constant in the time, (making the first integral zero) and then using in-terpolation estimate (17) in the second integral. It re-mains to estimate the terms II, to this end, we need the following notation: nnn  0R  ,,ddntntxxtx t so that:  ,d,ddnnntnnIIIkx xttxt  (21) where,  and ˆˆ.,nnt : 1010101120[]ˆ,[]wˆ,ˆ,[],:NhnnnnNhnnnnnNhnnnnnNhnnnnwIIk kkPPkwkPkwkPII IIk To estimate 1II, we use (21) to get:  1221221110110nnnnRkPk1101102() ()1() ()ˆ,ˆˆ,.,d.,dd.,d dnnnnnNnn nnNnNtnn IItnNtIt nnntLLntLLIIkRPRkPt tPtRP tkR PkR  As for the n 2II-terms we can write: 121201,,01,,01,0,.nnInnPIk1,02()[],,(.,d.,d d,d,.,ddnnnnnNhnnnnnnNhhnnnnnnnNnhh nInntIt nnNnhnnNnhnIInnnnLwIIk PkwwPIkkPw wPI ttktPIw t tPIw PIt tkkR12222()() ()xx ntLLLkR The a posteriori error estimate now follows immedi-ately after collecting the terms and using the definition of the stability factors (18) and (19). 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