1. 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 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:
(1)
where
is a bounded open set in
is a scalar-valued function representing the unknown,
is a finite time,
, n is the unit outward normal to the domain boundary,
is the advective velocity,
,
and
is the initial datum.
Let us introduce some notations and assumptions:
• Assume
is a Lipschitz continuous functions i.e.
where
denotes the Euclidean norm of
in
.
• We set
and
.
• We set
.
• Assume
.
•
,
and
.
• Let
, we consider the space
(2)
where V is a Hilbert space and equipped with the scalar product defined by:
(3)
The associated norm is:
[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.
2. Univariate B-Splines
Definition 1. Let
be an increasing sequence of reals, B-splines functions of degree k are defined by Cox-de Boor-Mansfield recursion formula [15] :
(5)
(6)
with the convention
for all real number x.
The set
is called knots vector.
Now, we want to look into bivariate B-splines, obtained from univariate B-splines.
3. Bivariate B-Splines and Geometry of the Physical Domain
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
. Mimicking the one-dimensional case, given integers
and
for
. Let us introduce open knot vectors:
and the associated vectors without repetitions for each direction l
There is a parametric cartesian mesh
associated with these knot vectors partitioning the parametric domain
into a rectangular grid. So, we have:
[16] (7)
For each element
, we associate a parametric mesh size
where
denotes the length of the largest edge of Q. Also, for each element, we define a shape regularity constant as in [16] :
(8)
where
denotes the length of the smallest edge of Q.
We associate with each knot vector
univariate B-spline basis functions
of degree
for
.
On the mesh
, we define the tensor-product B-spline basis functions as in [16] by:
(9)
(10)
The span of these functions form the space of two-dimensional splines over
, denoted by:
The physical domain
is defined through a geometrical mapping:
[16]
where
are the so-called control points. F is a parametrization of the physical domain
, that is,
For each element Q in the parametric domain
, there is a corresponding physical element
, as shown in Figure 1.
We assume throughout that F is invertible, with smooth inverse
, on each element
.
We define the physical mesh to be:
(11)
We assume (
) is quasi-uniform:
(12)
with
the diameter of K and
.
We introduce
, the space spanned by B-splines basis functions in
as the push-forward of the B-splines space
.
Given a function
, we define a projective operator over the B-splines space
as:
,
where the linear functionals
determine the dual basis for the set of B-splines.
The projective operator over the B-splines space
, is defined as the push-forward of the operator
.
4. Main Results
This section is devoted to our convergence results obtained for respectively a polynomial of degree
and a polynomial of degree
. We present our main results whose proofs are given in the subsection 6.6.
Theorem 1. (Convergence for RK2,
)
Assume the
CFL Condition:
Figure 1. Definition of domains used in isogeometric analysis (Source [17] ) (Page 181).
for some positive real number
(13)
and
for
. Then,
(14)
with
(15)
(16)
(17)
where
(18)
and
is the time step.
Theorem 2. (Convergence for RK2,
)
Assume the
CFL Condition, assume
for
and
. Then,
(19)
with
(20)
and
(21)
5. Inverse and Trace Inequalities
In this section, we present isogeometric inverse and trace inequalities, useful tools to analyze partial differential equations.
Let
and
.
Theorem 3. (see [11] [18] )
and
(22)
where
depends on k and on the parametrization F.
Theorem 4. (see [16] )
(23)
where C depends only on
and
,
is the local shape regularity constant of Q, and
is the shape regularity constant of K.
We set
(see. [4] [19] ).
Theorem 5. (see [20] ) Given the integers l and s such that
and a function
, then:
(24)
where C is independent on h.
Theorem 6. Given the integer s such that
and a function
, then:
(25)
where C is independent on h.
Proof 1. Let
. Using the inequality (23), we have:
(26)
being quasi-uniform,
.
(27)
So
(28)
Using the inequality (24), we have:
(29)
Thus, we get:
(30)
6. Functional Setting and Space Semidiscretization
6.1. Functional Setting
In this part, we introduce some basic notations for space-time functions and important theorems.
Theorem 7. (see. [11] )
is a Banach space when equipped with the norm:
(31)
with
[11] (page69) (32)
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
of functions in the space defined by:
[11] (33)
6.2. Space Semidiscretization
Considering (
), we present following notations:
• Interfaces are collected in the set
and boundary faces are collected in the set
. We set
.
.
•
, the mean of v is denoted by {{v}}.
• The jump of v is denoted by [v].
• Assume
is a partition of
such that, for the exact solution u,
[11]
where
(34)
We set
with
(35)
We define the discrete operator
such as
,
[11] (36)
6.3. Assumptions
For all
, set:
(37)
(38)
(39)
(40)
We abbreviate as
the inequality
with positive C independent of
. The value of C can change at each occurrence [11] .
We now state some assumptions on the discrete operator
. The first one (41) is important to introduce the notion of numerical fluxes:
1) For all
,
[11] (41)
2) From equality (41), Cauchy-Schwarz inequality and inverse inequality (22), we can infer:
For all
,
(42)
3) The three next assumptions are useful to bound the operator
.
For all
with
(43)
For all
(44)
For all
(45)
4) The two next inequalities are bounds of
and
.
(46)
(47)
5) The two last inequalities are obtained thanks to CFL condition and isogeometric inverse and trace inequalities:
(48)
(49)
For the time discretization, we are interested in an explicit scheme: the two stage Heun scheme.
7. The Explicit Two Stage Heun Scheme Analysis
In this section, we want to tackle the convergence analysis of the two stage Heun scheme.
7.1. The Explicit Two Stage Heun Scheme
Let
be the time step such as
where N is an integer. For
, we define the discrete times
and
. Assume
with
.
We consider the following explicit scheme:
[11]
7.2. Error Equation
This step is to identify the error equation governing the time evolution of
and
.
We set
(50)
(51)
(52)
(53)
with
[11] (54)
From (50) and (51), we have
.
From (52) and (53), we have
.
We get:
[11] (55)
[11] (56)
where
(57)
where
(58)
and
(59)
7.3. Energy Identity
This step is to derive an energy identity for our scheme (6.1).
[11] (60)
with
(61)
7.4. 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
(68). To get it, we need the next lemma.
Lemma 1. Let
denote the L2-orthogonal projection onto
.
is spanned by piecewise constant functions on
.
Then,
,
(62)
where
is independent of
and of
.
Proof 2. This result is obtained using Cauchy-Schwarz inequality and equality (41).
7.5. Preliminary Results
This lemma is a preliminary stability bound.
Lemma 2. Assume
.
Assume the CFL condition:
for some positive real number
. (63)
Thus,
(64)
where C is independent of
and
.
Proof 3. Using CFL condition, energy identity (60), inequalities (46) and (47), we get (64).
Lemma 3. (Stability lemma,
) Assume
.
Assume the (
CFL Condition)
for some positive real number
(65)
Then, we infer:
(66)
Proof 4. The stability lemma (66) for a polynomial of degree
is obtained by bounding the term
in the energy identity (60).
Lemma 4. (Stability lemma,
) Assume
.
Assume the CFL condition:
for some positive real number
with
(67)
Thus,
(68)
where C is independent of
and
.
Proof 5. This lemma is proven as in [11] (Page 96).
7.6. Proofs of Our Main Results
Proof of theorem 1
Using the triangle and Young inequalities, we deduce:
Thus, we obtain:
(69)
Let
Set
Given
From the relation (66), we deduce that:
Set
(70)
whence, we have:
(71)
Applying the Gronwall lemma, we get for
:
(72)
So
for
(73)
So
(74)
where
Therefore, we have:
(75)
From inequalities (69), (74) and (75), we get:
(76)
From inequalities (48), (49) and CFL condition, we obtain:
Using inverse inequality (24),
So
Therefore, inequality (76) becomes:
(77)
Set
and
(78)
We obtain thus:
Proof of theorem 2
Using inequality (68), we get inequality:
(79)
with
Thus
So
(80)
whence
(81)
Therefore,
(82)
(83)
(84)
Remark 1. When F is the identity mapping,
so
[16] (page 10). Therefore
. We get same results as Alexandre Ern. Thus, Our results are a generalization of Ern results because in finite elements method, Ern obtained his error estimate, working on a polygon [11] . In the framework of our work, we got the same order of precision in time and space like Alexandre Ern. But our result is obtained for anygeometry.
8. Conclusion
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.
Acknowledgements
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.