A Posteriori Error Estimate for Streamline Diffusion Method in Solving a Hyperbolic Equation

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 increasingly powerful; such that there are various approaches to a posteriori error estimates and it has new successfully applied to varied problems by several authors (see Ainsworth and Tinsley Oden [1]; Asadzadeh [2]; Gergouli [3]; Johnson [4] and [5]). Gergouli et al. [3] and his teammates applied finite element method for linear wave equation and obtained a posteriori error estimates in L (L) 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 estimates. 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 [6]; Asadzadeh and Kowalczyk [7]; Eriksson and Johnson [8]; Brenner [9]; Dubois [10]; Fuhrer [11] ) 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 characteristic 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 [12]; Haws, [13]; Gergoulus et al., [3]; Iraniparast [14]; Kalmenov [15]; in Sobolov space Adams [16]; Shermenew [17]) as follows:


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 increasingly powerful; such that there are various approaches to a posteriori error estimates and it has new successfully applied to varied problems by several authors (see Ainsworth and Tinsley Oden [1]; Asadzadeh [2]; Gergouli [3]; Johnson [4] and [5]).
[3] and his teammates applied finite element method for linear wave equation and obtained a posteriori error estimates in L (L 2 ) 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 estimates.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 [6]; Asadzadeh and Kowalczyk [7]; Eriksson and Johnson [8]; Brenner [9]; Dubois [10]; Fuhrer [11] ) 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 characteristic direction so that internal layers are not smeared out, while the added diffusion removes oscillations near boundary layers.
In order to make use of the theory of Semigroups we write the system (1) in the following abstract form: Here, we assume for and where, I is identity matrix.
The rest of this study is organized as follows.In Section 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 interpolation estimates for dual problem.In Section 5, we complete proof for a posteriori error estimates by using definitions in Section 4.

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 0 1 be a subdivision of the time interval into intervals , with time steps 1 n n n , and introduce the corresponding space-time slabs, i.e.: for .Further, for each n let be a finite element subspace of 0,1, , , (see Adams, [16]) and let: We can formulate SD-method on the slab for (2), as follows: where, Ch

 
with C is a suitable chosen (sufficiently small, see Johnson, [18]) positive constant and parameter h is defined in the following.Further, we de-fine the following notations for (6): The terms including ,   in the above formula is a jump conditions which imposes a weakly enforced continuity condition across the slab interfaces, at t n 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: where, we define such that for , ,  For ,we define such that be a triangulation of the slab n into triangles K satisfying as usual the minimum angle condition (Ciarlet [21]) and assume that the parameter h is represented with the maximum diameter of the triangles

 
k P K denotes the set of polynomials in K of degree less than or equal k and: Thus (6) can be formulated as follows: Moreover, we know that the exact solution of (6) satisfies: and by use ( 6) and ( 7), we have the Galerkin ogon orth ality relation: where, . An a Posteriori Error Estimate for the this section, we shall consider the following simplified where, h e w w   .

SD-Method
In version of SD-method for (7) with  = 0: Find w h W h , such that for 0,1, , mplicit take 0 , 0 h w   .For si y, we 0 0 w  and .In order to the error, we i n e (10) and denotes the adjoint of the operator L defined in nd 0 F  con obtain a representation of sider the following auxiliary problem, referred to as the linearized dual problem: Find  such that: (2) a  is a positive weight function.Note that this problem is computed "backward", but there is a corresponding change in sign.Further, we shall first introduce the following notation: Multiplying ( 10) by e and integrating by parts and summing over n, we obtain the following error representation formula: e A e eA We have for by part integrating: d We define and 0,1, , We define: According to (9), Then in (12), by use ( 15) and ( 16), we have: So that recalling (9) and using the Galerkin orthogonality (8), we obtain: where : in space and in space time, respectively, by: by letting: where, . Further, if we introduce P and  de- respectively, then we can let to Now, we define residual of computed solutio by: n ˆh W   be: , ,on where, I is the identity operator.
In the end of this section, we shall give a lemma for projection operators P, le interpolation estimates by the aving the overall of I and II to next section.Lemma 1: There is a constant C such that for residual Proof: (see Johnson and Szepessy [19] and Sa [20]).

Completion of the Proof of a
n osteriori error estiate by estimating of the terms I and II in the error rep-ndboge

The Posteriori Error Estimates
this section we state and prove a p I m resentation formula (16).To this approach we introduce the stability factors (see Burman [18]) associated with discretization in time and space, defined by: respectively.We now apply the result of the previous sections; using Catchy-Schwartz inequality in ( 16) coupled with the interpolation estimate (17) and the strong stability factors (18) and (19), to derive the L L a posteriori error estimates for the scheme (9).
Theorem 1: The error h e w w   , where w solution of the continuous problems (2) and w is the that of (9), sa h tisfies the following stab te: ility estima Proof: Using the notation introduce above, we write (17) as: Below we shall estimate the terms I and II separately.


Copyright © 2011 SciRes.AM Splitting the interpolation error by writing and , we have: where we have used the fact that is constant in the time, (making the first integral zero) and then using interpolation estimate (17) in the second integral.It remains to estimate the terms II, to this end, we need the following notation: where, : To estimate 1 II , we use (21) to get: .
The a posteriori error estimate now follows immediately after collecting the terms and using the definition of the stability factors (18) and (19).
For 0   in (7), we can obtain a posteriori error estimates with similar way.
d    is a bounded open polygonal domain with boundary  and we have For g w   and where the bilinear form   ., .B and the linear form   .L 1 N  define by:

2 II
-terms we can write: