Full Discretization to an Hyperbolic Equation with Nonlocal Coefficient

Manal Djaghout, Abderrazak Chaoui and Khaled Zennir abstract: We present full discretization of the telegraph equation with nonlocal coefficient using Rothefinite element method. For solving the equation numerically we use the Newton Raphson method, but the nonlocal term causes difficulties because the Jacobien matrix is full. To remedy these difficulties we apply the technique used by Sudhakar [4]. The optimal a priori error estimates for both semi discrete and fully discrete schemes are derived in V , introduced in (1.4), and H(Ω) and a numerical experiment is described to support our theoretical result.


Introduction and Preliminaries
Let Ω is a simply connected bounded domain of R d , d ≥ 2 with Lipschitz continuous boundary ∂Ω. Consider the following nonlocal hyperbolic problem We introduce the elliptic differential operator A defined by where A(x) is a symmetric matrix with entries that are uniformly bounded and measurable, b(x) is a bounded positive function and we assume that f , u 0 , u 1 and A(x) are smooth enough functions. The acoustic telegraph equation (1.1) with nonlocal term and constant coefficients is used to model the effects of diffusion and wave propagation by introducing a term that accounts for effects of finite velocity to standard heat or mass transport equation (see [1]). The function a in equation (1.1) is the diffusion depends an a nonlocal quantity Ω u(x, t)dx and assumed to depend on the entire population in the domain Ω. Recent years have seen an increasing interest in studying nonlocal problems, of this type of problems [ [4], [5], [8]]. One of the more popular methods for solving partial differential equation is the Roth method (or the 2 M. Djaghout, A. Chaoui and Kh. Zennir method of lines), this method is used in the time discretization of evolution equations where the derivatives with respect to one variable are replaced by the corresponding difference quotients that finally leads to systems of differential equations for functions of the remaining variables. Roth's method was introduced by E. Roth in his the pioneer work1930, it has been adopted and developed by many authors for example O.A. Ladyzenskaja [9], [10] and K. Rektorys [ [13], [14]] for solving second order linear and quasilinear parabolic problems. Recently Roth's method has been studied linear and quasilinear hyperbolic equations [ [6], [15], [3]]. The purpose of this work is to combine Rothe's method with finite element. The fully discrete scheme for problem (1.1) gives a system of nonlinear equations, we use Newton Raphson method to solve this system. It is known that the Newton Raphson iteration is the most popular for solving nonlinear algebraic equations because it is fast convergent in a small number of iteration. One of the main difficulties of using Newton's is the fully Jacobien matrix, this difficulty can be addressed by reformulate the system as [4]. The paper is organized as follows : In section 1, we present some basic notations needed material. Section 2 contains the weak formulation, the discretization scheme based on Rothe's method and a priori estimates. In section 3 we give the fully discrete scheme and a priori error estimates. Finally, a numerical example is presented in section 4. Let (., .) denote the inner product in L 2 (Ω), and let (., .) A be the inner product of 5) and the norms on L 2 (Ω), V are denoted . , . A respectively. We take C ǫ = C(ǫ −1 ) with ǫ is small. For m ≥ 0, we use H m (Ω) to denote the Sobolev space on Ω of order m with the norm Along this work we shall always assume the following assumptions: and satisfies the condition of growth 3. (H3) a : R → R is Lipschitz continuous with the Lipschitz constant L M , this means Full Discretization to an Hyperbolic Equation

3
Definition 1.1. A function u is said a weak solution of (1.1) if

Time Discretization
We divide the interval [0, T ] into n subintervals of length τ = T n and denote u i = u(t i , x), t i = iτ , i = 0, 1, ..., n. Let u −1 be defined as the recurrent approximation scheme for i = 1, ..., n becomes .., n, such that, We define the Roth's functions by a piecewise linear interpolation with respect to the time t, together with the step functionū We denote byf n the functionf Then, the problem (2.1) can be takes the form: (2.6) By integrating the above equation over [0, T ], we get Using Young, we obtain Taking summation from i = 1 to s, we get Applying the Abel's summing formula, we obtain Using the Gronwall's Lemma (see, e.g. [11]) inequality and choosing ǫ = τ to get We denote by e u = u − u n and e f = f −f n .

. Full Discretization
At each time t i , 0 ≤ i ≤ n, we consider a triangulation Υ i h made of triangles T i such that no nodes of every triangle lies in the interior of a side of another triangle. Let V i h be the discrete space of V i defined by be the basic functions for the space V i h such that any function will be the pyramid form in V i h and which takes the value 1 at {p j } N j=1 and vanishes at exterior nodes. We can write the solution u i h as Let X be a Banach space, we use the following norm in discrete version.
Then, the fully discrete scheme for problem (1.1) reads as We introduce the orthogonal projection operator Π i h : From fully discrete weak formulation of (3.2), we have and, ∀v ∈ V i h , The problem (3.5) give as a system of nonlinear algebraic equations by using finite element, then can be given this system as follows : We use Newton-Raphson method to solve (3.5), but the presence of nonlocal term in the equation destroys the sparsity of Newton-Raphson method. We compute the Jacobian matrix J To get the value of α i j by Newton's method, every element of the Jacobian matrix takes the form In order to ensure the sparsity of the Jacobian matrix we modify the scheme (3.5) according to the technic used by Chaudhary in [4]. Then the problem (3.5) can be rewritten as follows : Take v h = φ j , and reformulate the equations (3.9)-(3.10) as follows: T . The matrix system (3.12) can be solved by using the Sherman-Morrison Woodbury formula and block elimination with one-refinement algorithm in [8], [7]. We introduce the orthogonal projection to get an optimal convergence between u i , u i h . Therefor, we can take the error as follows. Choosing Thus, Using Poincaré inequality, we obtain where c and C are some positive constants.
We use the same proof in Lemma 2.1 to obtain the existence of u i h and a priori estimates.
This means We integrate from 0 to T , to obtain  where c is given in Eq. (3.18). Then, there exists a positive constant C such that Proof : From equations (2.1), (3.1), we have Thus, (3.21) Choosing v h = τ 2 δθ i h in (3.21), we obtain New left-hand side of (3.22) can be estimated as follows.
To estimate the right-hand side of (3.22), we need the following steps.
Step1. We estimate − τ 2 ∂ t δρ i h , δθ i h − τ 2 ∂ t ρ i h , δθ i h . Using Cauchy-schwarz, we get Full Discretization to an Hyperbolic Equation

Numerical experiment
In this section, we set up a numerical experiment to find an approximate solution of problem (1.1), if we use Roth's approximation in time discretization and finite element scheme in the spatial discretization in which we prescribe the computational domain Ω = (0, 1), the time interval (0, 1) i.e. T = 1 and we take A(x) = b(x) = 1.
In order using Newton's we take initial guess u 0 and u 1 as follows u 0 = 0, and u 1 =    1, at interior node 0, at boundary node The tolerance for stopping iteration is defined to be 10 −15 , we have considered the step length h = 1 10 , 1 20 , 1 30 , 1 40 and τ = 0.001. We plot the error in log log-plot. We choose f (x, t, u) according to test solution u(x, t) = x(1 − x)te −t 2 and a(l(u)) = 1 + cos(l(u)). The table below gives the numerical errors.