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 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 h- and 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:

{ ∂ t u ( x , t ) + ∇ ⋅ ( β ( x ) u ( x , t ) ) = 0 , x ∈ Ω ⊂ ℝ 2 , t ∈ [ 0 ; t f ] u ( . , t = 0 ) = u 0 u = 0     in   ∂ Ω − × [ 0 ; t f ] (1)

where Ω is a bounded open set in ℝ 2 , u : Ω × [ 0 ; t f ] ↦ R 2 is a scalar-valued function representing the unknown, t f is a finite time, ∂ Ω − = { x ∈ ∂ Ω , β ( x ) ⋅ n ( x ) < 0 } , n is the unit outward normal to the domain boundary, β is the advective velocity, β ∈ [ L ∞ ( Ω ) ] 2 , ∇ ⋅ β ∈ L ∞ ( Ω ) and u 0 is the initial datum.

Let us introduce some notations and assumptions:

• Assume β is a Lipschitz continuous functions i.e.

∃ L β , ∀ ( x , y ) ∈ Ω 2 , ‖ β ( x ) − β ( y ) ‖ ≤ L β ‖ x − y ‖ .

where ‖ x − y ‖ denotes the Euclidean norm of ( x − y ) in R 2 .

• We set τ c : = 1 max { ‖ β ′ ‖ L ∞ ( Ω ) ; L β } and β c : = ‖ β ‖ [ L ∞ ( Ω ) ] 2 .

• We set τ ⋆ : = min ( t f , τ c ) .

• Assume h ≤ β c τ ⋆ .

• ∀ x ∈ ℝ , x ⊖ : = 1 2 ( | x | − x ) and x ⊕ : = 1 2 ( | x | + x ) .

• Let l ∈ ℕ , we consider the space

C l ( V ) : = C l ( [ 0, t f ] , V ) , (2)

where V is a Hilbert space and equipped with the scalar product defined by:

( u , s ) V = ( u , s ) L 2 ( Ω ) + ( ∇ ⋅ ( β u ) , ∇ ⋅ ( β s ) ) L 2 ( Ω ) , ∀ ( u , s ) ∈ V 2 . (3)

The associated norm is:

‖ u ‖ V 2 = ‖ u ‖ L 2 ( Ω ) 2 + ‖ ∇ ⋅ ( β u ) ‖ L 2 ( Ω ) 2 , ∀ u ∈ V . [11] (page 39) (4)

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.

Definition 1. Let x 1 ≤ x 2 ≤ ⋯ ≤ x m be an increasing sequence of reals, B-splines functions of degree k are defined by Cox-de Boor-Mansfield recursion formula [15] :

{ For   1 ≤ i ≤ m − 1 N i , 0 ( t ) = 1         if   t ∈ [ x i , x i + 1 [ N i , 0 ( t ) = 0             otherwise (5)

{ For     k ≥ 1   and   1 ≤ i ≤ m − k − 1 N i , k ( t ) = t − x i x i + k − x i N i , k − 1 ( t ) + x i + k + 1 − t x i + k + 1 − x i + 1 N i + 1 , k − 1 ( t ) , (6)

with the convention x 0 : = 0 for all real number x.

The set ( x i ) i = 1 m   ( 1 ≤ i ≤ m ) is called knots vector.

Now, we want to look into bivariate B-splines, obtained from univariate B-splines.

The definition of bivariate B-splines follows easily through a tensor-product construction. Let us focus on the two-dimensional case. Notably, let us consider the unit square Ω ^ = [ 0 ; 1 ] 2 ⊂ ℝ 2 . Mimicking the one-dimensional case, given integers p l and n l for l = 1 , 2 . Let us introduce open knot vectors:

E l = { ξ 1 , l , ⋯ , ξ n l + p l + 2 , l }

and the associated vectors without repetitions for each direction l

ζ l = { ζ 1 , l , ⋯ , ζ m l , l }

There is a parametric cartesian mesh Q h associated with these knot vectors partitioning the parametric domain Ω ^ into a rectangular grid. So, we have:

Q h = { Q = ⊗ l = 1 , 2 ( ζ i l , l , ζ i l + 1 , d ) , 1 ≤ i l ≤ m l − 1 } [16] (7)

For each element Q ∈ Q h , we associate a parametric mesh size h Q = h Q , max where h Q , max denotes the length of the largest edge of Q. Also, for each element, we define a shape regularity constant as in [16] :

λ Q = h Q h Q , min (8)

where h Q , min denotes the length of the smallest edge of Q.

We associate with each knot vector E l , ( l = 1 , 2 ) univariate B-spline basis functions N i , p l of degree p l for i = 1 , ⋯ , n l .

On the mesh Q h , we define the tensor-product B-spline basis functions as in [16] by:

N ( i , j ) , ( p 1 , p 2 ) = N i , p 1 ⊗ N j , p 2 , i = 1 , ⋯ , n 1 , j = 1 , ⋯ , n 2 . (9)

N ( i , j ) , ( p 1 , p 2 ) = N i , p 1 N j , p 2 , i = 1 , ⋯ , n 1 , j = 1 , ⋯ , n 2 . (10)

The span of these functions form the space of two-dimensional splines over Ω ^ , denoted by:

S h = s p a n { N ( i , j ) , ( p 1 , p 2 ) } i = 1 , j = 1 n 1 , n 2

The physical domain Ω is defined through a geometrical mapping:

F = ∑ i = 1 n 1 ∑ j = 1 n 2   P i j N i , n 1 N j , n 2 [16]

where P i j ∈ ℝ 2 are the so-called control points. F is a parametrization of the physical domain Ω , that is,

F : [ 0,1 ] 2 → Ω

For each element Q in the parametric domain [ 0,1 ] 2 , there is a corresponding physical element K = F ( Q ) , as shown in Figure 1.

We assume throughout that F is invertible, with smooth inverse F − 1 , on each element Q ∈ Q h .

We define the physical mesh to be:

τ h = { K : K = F ( Q ) , Q ∈ Q h } = F ( Q h ) . (11)

We assume ( τ h ) is quasi-uniform:

∃ C > 0 , h ≤ C h K . (12)

with h K the diameter of K and h : = max K ∈ τ h h K .

We introduce V h , the space spanned by B-splines basis functions in Ω as the push-forward of the B-splines space S h .

V h : = s p a n { N ( i , j ) , ( p 1 , p 2 ) ∘ F − 1 } i = 1 , j = 1 n 1 , n 2 .

Given a function v ^ ∈ L 2 ( Ω ^ ) , we define a projective operator over the B-splines space S h as:

π S h : L 2 ( Ω ^ ) → S h , π S h v ^ : = ∑ i = 1 , j = 1 n 1 , n 2 φ ( v ^ ) N ( i , j ) , ( p 1 , p 2 ) ,

where the linear functionals φ ∈ L 2 ( Ω ^ ) ′ determine the dual basis for the set of B-splines.

The projective operator over the B-splines space V h , is defined as the push-forward of the operator π S h .

π h : L 2 ( Ω ) → V h , π h v : = ( π S h ( v ^ ) ) ∘ F − 1

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

Theorem 1. (Convergence for RK2, k ≥ 2 )

Assume the 4 3 CFL Condition:

δ t ≤ ϱ ′ τ ⋆ − 1 3 ( h β c ) 4 3 for some positive real number ϱ ′ (13)

and d t s u ∈ C 0 ( H k + 1 − s ( Ω ) ) for s ∈ { 0,1 } . Then,

‖ u N − u h N ‖ L 2 ( Ω ) + ( ∑ m = 0 N − 1   δ t | u m − u h m | β 2 ) 1 2 ≲ exp ( 4 C r 8 τ ⋆ t f ) ( χ 1 δ t 2 + χ 2 h k + 1 2 ) (14)

with

χ 1 = r 4 t f 1 2 τ ⋆ 1 2 ‖ d t 3 u ‖ C 0 ( L 2 ( Ω ) ) (15)

χ 2 = t f 1 2 r 5 β c 1 2 ‖ u ‖ C 0 ( H k + 1 ( Ω ) ) + t f 1 2 r 5 β c − 1 2 ‖ d t u ‖ C 0 ( H k ( Ω ) ) (16)

| v | β 2 = ∫ ∂ Ω 1 2 | β ⋅ n | v 2 + ∑ F ∈ F h i ∫ F 1 2 | β ⋅ n F | [ v ] 2 , (17)

where

r = max ( 1 , max K ∈ τ h ( λ Q λ K ) ) (18)

and δ t is the time step.

Theorem 2. (Convergence for RK2, k = 1 )

Assume the 4 3 CFL Condition, assume d t s u ∈ C 0 ( H k + 1 − s ( Ω ) ) for s ∈ { 0,1 } and 4 − 3 r 2 > 0 . Then,

‖ u N − u h N ‖ L 2 ( Ω ) + ( ∑ m = 0 N − 1   δ t | u m − u h m | β 2 ) 1 2 ≲ 1 4 − 3 r 2 exp ( 8 C r 8 τ ⋆ t f ) ( χ 1 δ t 2 + χ 2 h k + 1 2 ) (19)

with

χ 1 = r 4 t f 1 2 τ ⋆ 1 2 ‖ d t 3 u ‖ C 0 ( L 2 ( Ω ) ) (20)

and

χ 2 = t f 1 2 r 5 β c 1 2 ‖ u ‖ C 0 ( H k + 1 ( Ω ) ) + t f 1 2 r 5 β c − 1 2 ‖ d t u ‖ C 0 ( H k ( Ω ) ) (21)

In this section, we present isogeometric inverse and trace inequalities, useful tools to analyze partial differential equations.

Let K ∈ τ h and Q = F − 1 ( K ) .

Theorem 3. (see [11] [18] )

∀ h > 0 , ∀ K ∈ τ h and ∀ v h ∈ ℙ 2 k ( τ h ) , ‖ ∇ v h ‖ [ L 2 ( K ) ] 2 ≤ C ⋆ h K − 1 ‖ v h ‖ L 2 ( K ) (22)

where C ⋆ depends on k and on the parametrization F.

Theorem 4. (see [16] )

∀ v h ∈ ℙ 2 k ( τ h ) , ‖ v h ‖ L 2 ( ∂ K ) 2 ≤ C λ Q λ K h K − 1 ‖ v h ‖ L 2 ( K ) 2 (23)

where C depends only on p 1 and p 2 , λ Q is the local shape regularity constant of Q, and λ K is the shape regularity constant of K.

We set p : = min { p 1 , p 2 } (see. [4] [19] ).

Theorem 5. (see [20] ) Given the integers l and s such that 0 ≤ l ≤ s ≤ p + 1 and a function u ∈ H s ( Ω ) , then:

∑ K ∈ τ h | u − π h u | H l ( K ) 2 ≤ C h 2 ( s − l ) ‖ u ‖ H s ( Ω ) 2 , (24)

where C is independent on h.

Theorem 6. Given the integer s such that 0 ≤ s ≤ p + 1 and a function u ∈ H s ( Ω ) , then:

∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( ∂ K ) 2 ≤ C max K ∈ τ h ( λ Q λ K ) h ( 2 s − 1 ) ‖ u ‖ H s ( Ω ) 2 , (25)

where C is independent on h.

Proof 1. Let u ∈ H s ( Ω ) . Using the inequality (23), we have:

∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( ∂ K ) 2 ≤ C ∑ K ∈ τ h λ Q λ K h K − 1 ‖ u − π h u ‖ L 2 ( K ) 2 , (26)

( τ h ) h being quasi-uniform, h ≤ C h K .

h ≤ C h K ⇒ C − 1 h K − 1 ≤ h − 1                           ⇒ h K − 1 ≤ C h − 1                           ⇒ C h K − 1 ≤ C ′ h − 1 (27)

So

∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( ∂ K ) 2 ≤ C max K ∈ τ h ( λ Q λ K ) h − 1 ∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( K ) 2 (28)

Using the inequality (24), we have:

∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( K ) 2 ≤ C h 2 s ‖ u ‖ H s ( Ω ) 2 (29)

Thus, we get:

∑ K ∈ τ h ‖ u − π h u ‖ L 2 ( ∂ K ) 2 ≤ C max K ∈ τ h ( λ Q λ K ) h ( 2 s − 1 ) ‖ u ‖ H s ( Ω ) 2 (30)

In this section, we want to tackle the convergence analysis of the two stage Heun scheme.

The isogeometric method has been used to establish an error estimate for transport equation in 2D using the explicit two stage Heun scheme, for smooth solutions, in the energy norm comprising the L²-norm and the jumps. These results generalize Ern results. An extension of this present paper is to tackle Burgers equation to get an isogeometric error estimate.

This article has been written in the framework of my Phd works, supported by CEA-SMA (Centre d’Excellence Africain en Sciences Mathématiques et Applications), funded by World Bank.

The authors declare that there is no conflict of interest as far as the publication of this paper is concerned.

Goudjo, A. and Aguemon, U. (2019) An Isogeometric Error Estimate for Transport Equation in 2D. Advances in Pure Mathematics, 9, 777-793. https://doi.org/10.4236/apm.2019.99037