An Isogeometric Error Estimate for Transport Equation in 2D

In this paper, an isogeometric error estimate for transport equation is obtained in 2D to prove the convergence of isogeometric method. The result that we have obtained, generalizes Ern result, got in finite elements method. For the time discretization, the two stage Heun scheme is used to prove this result. For a polynomial of degree 1 k ≥ , the order of convergence in space is 2 and in time is 1 2 k + .


Introduction
Some phenomena of the daily life such as particles transport in an electric field, the signal transport along a wire, evolution of cars on a road [1], and evolution of a pollutant in a narrow channel [2] are modelled by a transport equation. Study of numerical methods for solving this equation is very important to describe, to predict and to control these phenomena.
Isogeometric Analysis has been introduced by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005 [3].
The objectives of Isogeometric Analysis are to generalize and improve upon Finite Element Analysis (FEA) in the following ways: 1) To provide more accurate modeling of complex geometries and to exactly represent common engineering shapes such as circles, cylinders, spheres, ellipsoids, etc.
2) To fix exact geometries at the coarsest level of discretization and eliminate How to cite this paper: Goudjo, A. and geometrical errors.
3) To vastly simplify mesh refinement of complex industrial geometries by eliminating the necessity to communicate with the CAD (Computer Aided Design) description of geometry. 4) To provide refinement procedures, including classical hand p-refinements analogues, and to develop a new refinement procedure called k-refinement [4].
The idea of Isogeometric Analysis is to build a geometry model and, rather than develop a finite element model approximating the geometry, directly use the functions describing the geometry in analysis [5] [6]. These functions are B-splines.
Isogeometric Analysis is approached, using continuous or discontinuous Galerkin method. In the context of space semidiscretization by discontinuous Galerkin methods, explicit RK schemes are used to approximate in time systems of ordinary differential equations. These schemes have been developed by Cockburn and Shu [7], Cockburn, Lin, and Shu [8], and Cockburn, Hou, and Shu [9] and applied to a wide range of engineering problems [10]. They have been used by Alexandre Ern et al. [11] [12], for linear conservation laws using Discontinuous Galerkin Method to prove a convergence result [12]. Authors did a space semidiscretization using the upwind DG method. Besides, others tools are fundamental to get this convergence result: 1) Error equation.
2) An energy identity obtained from error equations.
3) A stability estimate using Gronwall lemma, Young inequality and inverse and trace inequalities for finite elements method.
In the literature, there exist many numerical methods to solve transport equation [13] [14]. To our best knowledge, there is no error estimate for transport equation using isogeometric method. In our work, we give such an estimate to generalize results obtained by Alexandre Ern et al. [11] [12] in finite elements. In the framework of this dissertation, we want to prove a convergence result using isogeometric method. Among others, unlike finite elements, as far as the space semidiscretization is concerned, we have: 1) Constructed a parametrization of the physical domain, indispensable to describe this domain.
2) Constructed a parametric mesh making a tensor product of knot vectors.
3) Introduced the discrete space on the physical domain, using our parametrization.
Moreover, instead of using inverse and trace inequalities for finite elements method, we will use isogeometric inverse and trace inequalities to obtain our convergence result. As far as the discretization in time is concerned, the explicit two stage Heun scheme is used. Now, we consider the following model: ., 0 x n x β − ∂Ω = ∈ ∂Ω ⋅ < , n is the unit outward normal to the domain boundary, β is the advective velocity, Let us introduce some notations and assumptions: • Assume β is a Lipschitz continuous functions i.e.
denotes the Euclidean norm of ( ) • We set

( )
: min , • Let l ∈  , we consider the space where V is a Hilbert space and equipped with the scalar product defined by: The associated norm is: This paper is organized as follows. In the first section, we will describe univariate B-splines. In the second one, we will describe bivariate B-splines and geometry of the physical domain. In the third one, we present main results of this work. In the fourth one, we will state inverse and isogeometric inequalities. In the fifth one, we will talk about the functional setting and space semidiscretization. In the sixth one, we will look into the explicit two stage Heun scheme analysis.

Univariate B-Splines
be an increasing sequence of reals, B-splines functions of degree k are defined by Cox-de Boor-Mansfield recursion formula [15]: For each element h Q Q ∈ , we associate a parametric mesh size denotes the length of the largest edge of Q. Also, for each element, we define a shape regularity constant as in [16]: On the mesh h Q , we define the tensor-product B-spline basis functions as in [16] by: The span of these functions form the space of two-dimensional splines over Ω , denoted by:  [16] where 2 ij P ∈  are the so-called control points. F is a parametrization of the physical domain Ω , that is, , as shown in Figure 1.
We assume throughout that F is invertible, with smooth inverse We define the physical mesh to be: with K h the diameter of K and : max We introduce h V , the space spanned by B-splines basis functions in Ω as Ω , we define a projective operator over the B-splines space h S as: ( ) ( : , :

Main Results
This section is devoted to our convergence results obtained for respectively a polynomial of degree 2 k ≥ and a polynomial of degree 1 k = . We present our main results whose proofs are given in the subsection 6.6.
and t δ is the time step.

Inverse and Trace Inequalities
In this section, we present isogeometric inverse and trace inequalities, useful tools to analyze partial differential equations.
Theorem 3. (see [11] [18]) 0, where C depends only on 1 p and 2 p , Q λ is the local shape regularity constant of Q, and K λ is the shape regularity constant of K.
We set Theorem 5. (see [20]) Given the integers l and s such that 0 1 l s p ≤ ≤ ≤ + and a function ( ) s u H ∈ Ω , then: where C is independent on h. Theorem 6. Given the integer s such that 0 Using the inequality (23), we have: Using the inequality (24), we have: Thus, we get:

Functional Setting
In this part, we introduce some basic notations for space-time functions and important theorems.
We want to specify mathematically the meaning of the boundary condition 1.
Our aim is to give a meaning to such traces in the space. Thus, we need to investigate the trace on ∂Ω of functions in the space defined by:

Space Semidiscretization
Considering ( h τ ), we present following notations: • Interfaces are collected in the set = Ω is a partition of Ω such that, for the exact solution u, We set We define the discrete operator :

Assumptions
For all h v V ∈  , set: We abbreviate as a b  the inequality a Cb ≤ with positive C independent of , , h t β δ . The value of C can change at each occurrence [11].
We now state some assumptions on the discrete operator h A . The first one (41) is important to introduce the notion of numerical fluxes: 3) The three next assumptions are useful to bound the operator h A .
For all 4) The two next inequalities are bounds of ( ) ( )

The Explicit Two Stage Heun Scheme Analysis
In this section, we want to tackle the convergence analysis of the two stage Heun scheme.

Energy Identity
This step is to derive an energy identity for our scheme (6.1).

Stability Estimate
Our aim is to bound the right terms in the energy identity (60).
We want now to establish a stability lemma for a polynomial of degree 1 k = (68). To get it, we need the next lemma.
where C   is independent of , h t δ and of β . Proof 2. This result is obtained using Cauchy-Schwarz inequality and equality (41).

Preliminary Results
This lemma is a preliminary stability bound.
Assume the CFL condition: Thus, Assume the ( 4 3 CFL Condition) Then, we infer: Assume the CFL condition: where C is independent of , h t δ and β .

Proofs of Our Main Results
Proof of theorem 1 Using the triangle and Young inequalities, we deduce: Thus, we obtain: Therefore, we have: From inequalities (69), (74) and (75), we get: From inequalities (48), (49) and CFL condition, we obtain: