EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 404-417
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Isogeometric Analysis approximation of linear elliptic
equations with L1 data

Yibour Corentin Bassonon1, Arouna Ouédraogo1,∗

1 Département de Mathématiques, Laboratoire de Mathématiques,
Informatique et Applications (L@MIA), Université Norbert ZONGO, Koudougou, Burkina
Faso

Abstract. Isogeometric Analysis (IgA) is a recent technique for the discretization of Partial Dif-
ferential Equations (PDEs). The main feature of the method is the ability to maintain the same
exact description of the computational geometry domain throughout the analysis process, includ-
ing refinement.
In the present paper, we consider, in dimension d ≥ 2 the Isogeometric Analysis approximation of
second order elliptic equations in divergence form with right-hand side in L1. We assume that the
family of meshes is shape regular and satisfies the discrete maximum principle. When the right-
hand side belongs to L1(Ω), we prove that the unique solution of the discrete problem converges to

the unique renormalized solution in W 1,q
0 (Ω), 1 ≤ q <

d

d− 1
. We also prove some error estimates

and include numerical tests for data with low smoothness.

2020 Mathematics Subject Classifications: 65N30, 35J25

Key Words and Phrases: Isogeometric analysis, NURBS approximation, L1 data, renormalized
solution

1. Introduction

This paper is devoted to the Isogeometric Analysis approximation of second order
linear elliptic equations in divergence form with L1-data. We study the following problem{

−div (A∇u) = f in Ω,
u = 0 on ∂Ω,

(1)

where Ω is an open, bounded and Lipschitz set of Rd, with d = 2 or d = 3, A is a coercive
matrix with coefficients in L∞(Ω) and f belongs to L1(Ω).
This problem frequently appears in applied sciences, being one of the basic problems

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4674

Email addresses: corentinbassonon@gmail.com (Y. C. Bassonon), arounaoued2002@yahoo.fr (A.
Ouédraogo)

https://www.ejpam.com 404 © 2023 EJPAM All rights reserved.



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 405

in mathematical fluid mechanics (see [7, 9]). For this class of problems, the maximum
principle is required to obtain physically admissible solutions.
The problem (1) has been studied in [3] by the standard Finite Element Method (FEM).

The authors proved that the discrete solution converges in W 1,q
0 (Ω), 1 ≤ q <

d

d− 1
to

the unique renormalized solution (see [6] for existence and uniqueness of renormalized
solution).
The Isogeometric Analysis based on NURBS (Non-Uniform Rational B-Splines), which
possesses improved properties, is a generalization of classical Finite Element Method.
NURBS are capable of more precise geometric representation of complex objects and can
exactly represent many engineered shapes. IgA also simplifies mesh refinement because
the geometry is fixed at the coarsest level of refinement and is unchanged throughout the
refinement process.
The rest of the paper is organized as follows: in Section 2, we gives setting of the problem
and main result. We recall brievly the Isogeometric Analysis method. In Section 3, we
study the convergence analysis and we obtain the error estimates for data in Lr,∞(Ω) for
1 < r < 2. To finish, we give numerical result. The novelty in our work is the convergence
result in W 1,q

0 (Ω), obtained for approximate solutions of (1) in NURBS space.

2. Preliminary

2.1. Renormalized solution

We investigate the Poisson’s problem with homogeneous boundary conditions under
the following conditions : the matrix A is such that

A ∈ L∞(Ω)d×d, (2)

a.e. x ∈ Ω, ∀ϕ ∈ Rd, A(x)ϕ · ϕ ≥ α|ϕ|2, (3)

for some α > 0, and the right-hand side f is such that

f ∈ L1(Ω). (4)

We give the definition of the renormalized solution of problem(1).

Definition 2.1. A function u is a renormalized solution of (1) if u satisfies

u ∈ L1(Ω), (5)

∀k > 0, Tk(u) ∈ H1
0 (Ω), (6)

lim
k−→∞

1

k

∫
Ω
|∇Tk(u)|2dx = 0, (7)

∀k > 0, ∀S ∈ C1
c (R) with suppS ⊂ [−k,+k],

∀v ∈ H1
0 (Ω) ∩ L∞(Ω),∫

Ω
A∇Tk(u)∇vS(u)dx+

∫
Ω
A∇Tk(u)∇Tk(u)S

′(u)vdx =

∫
Ω
fS(u)vdx.

(8)



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 406

As Tk(u) ∈ H1
0 (Ω), every term makes sense in (8).

When f belongs to L1(Ω) ∩H−1(Ω), the usual weak solution of (1), namely
∀u ∈ H1

0 (Ω),

∀v ∈ H1
0 (Ω),

∫
Ω
A∇u∇vdx =

∫
Ω
fvdx,

(9)

is also a renormalized solution of (1) and conversly.

2.2. NURBS-based Isogeometric Analysis

Here, we recall the basic concepts of the B-Splines and NURBS basis functions and
geometrical representation.
NURBS are built from B-Splines. A knot vector in one dimenion is a set of coordi-

nates in the parametric space, written Ξ =
{
ξ1, ξ2, ..., ξn+p+1

}
, where ξi is the i -knot

index i ∈ {1, ..., n + p + 1} characterized by the polynomial degree p and the number of
basis functions n defining the B-Splines basis, respectively. By convention, we assume
that ξ1 = 0 and ξn+p+1 = 1. The consequence is that parametric domain is defined as

Ω̂ := (ξ1, ξn+p+1) = (0, 1) ⊂ R. Knots may be repeated with the number of repetitions
indicating its multiplicity. To investigate the concept of mesh elements in the parametric
domain, we collect all the r distinct and ordered knots of Ξ, say ζj for j = 1, ..., r into
a vector Z = {ζ1, ..., ζr} with ζ1 ≡ ξ1 = 0 and ζr ≡ ξn+p+1 = 1. In particular, the one

dimensional mesh over Ω̂, say Qh, is given by

Qh := {Q = (ζj , ζj+1) : j = 1, ..., r − 1};

We denote by

h := max{hQ : Q ∈ Qh}, where hQ := diam(Q) ∀Q ∈ Qh (10)

the global mesh size in the parametric domain Ω̂.
By means of the Cox-de Boor recursion formula, (see [5], [10]), univariate B-Splines basis
functions Ni : Ω̂ −→ R for i = 1, ..., n, are built as piecewise polynomials of degree p
with compact support over the interval (ξi, ξi+p+1). The basis functions are everywhere
pointwise nonnegative and C∞−continuous, except in the knot values ζj , where they are
only Cp−mj− continuous. In particular, we define for all j = 1, ..., r, the smoothness
integer parameters kj = p − mj + 1 such that 0 ≤ kj ≤ p, we collect them in a vector
K = {k1, ..., kr}, and we introduce the minimum integer parameter kmin := min

j=2,...,r−1
{kj}.

The B-Splines space built from the basis function in the parametric domain Ω̂ reads:

Sh := span{Ni}ni=1. (11)

By definition, the B-Splines in Sh are globally Ckmin−continuous.



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 407

For each multi-index i := (i1, ..., iκ) in the set I = {i = (i1, ..., iκ) : 0 ≤ iα ≤
nα, for 1 ≤ α ≤ κ}, we define the multivariate B-Splines basis functions as:

Ni : Ω̂ → R, Ni(η) :=
κ∏

α=1

Nα
iα(ηα), (12)

and we denote the tensor product B-Splines space, as:

Sh := span{Ni}i∈I . (13)

Uni- and multivariate NURBS basis functions are defined on the parametric domain
Ω̂ = (0, 1)κ once provided κ knot vectors Ξα for α = 1, ..., κ and the corresponding
B-Splines basis {Ni}i∈I , by introducing a set of real numbers ω = {ωi}i∈I , called the
weights. We assume that the weights are positive and we define a positive scalar piecewise
polynomial function, called weighting function, as:

W : Ω̂ → R, W (η) :=
∑
i∈I

ωiNi(η). (14)

The i-th multivariate NURBS basis function is defined as

Ri : Ω̂ → R, Ri(η) =
Ni(η)ωi

W (η)
∀i ∈ I, (15)

and the corresponding NURBS space over the parametric domain Ω reads:

Nh := span{Ri}i∈I . (16)

Therefore, we consider the NURBS space over the parametric domain Ω̂ of (16) and a
set of control points {Pi}i∈I ⊂ Rd.Then a NURBS geometry Ω in Rd is defined from the
parametric domain Ω̂ = (0, 1)κ by means of the geometrical mapping :

x : Ω̂ → Ω ⊆ Rd x(η) =
∑
i∈I

Ri(η)Pi. (17)

By means of the geometrical mapping (17), we define the physical mesh Kh in the com-
putational domain Ω, whose elements are obatained as the image of the elements in the
parametric domain, i.e.:

Kh := {K = x(Q) : Q ∈ Qh}.

We denote the global mesh size of the mesh in the physical domain by

h := max{hK : K ∈ Kh}, with hK := ∥∇x∥L∞(K)ĥK̂ and ĥ
K̂

= diam(K̂).

Further, we assume that the physical mesh is quasi-uniform, i.e. there exists a positive
constant Cu, independent of h, such that

hK ≤ h ≤ CuhK ∀K ∈ Kh. (18)



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 408

Moreover, we define the space of NURBS in the domain Ω as the push-forward of the
space Nh of (16), i.e.:

Vh := span{Ri ◦ x−1}i∈I = span{Ri}i∈I , (19)

where {Ri}i∈I is the NURBS basis in the physical domain, with Ri := Ri ◦ x−1 for all
i ∈ I. The geometrical mapping (17) is assumed to be invertible a.e. in Ω, with smooth
inverse on each element K of the physical mesh Kh.
In our analysis, we restricted ourselves to the case d = κ.
In standard FEM, the space Vh is a space of piecewise polynomials. In an IgA context,
as introduced in [8], this space is formed by NURBS functions. For this, we introduce
finite-dimensional spaces on the patch (0, 1)d. The approximate solution uh of problem
(9) is obtained by solving the following problem:

Find uh ∈ V h,

∀vh ∈ V h,

∫
Ω
A∇uh∇vhdx =

∫
Ω
fvhdx,

(20)

where
V h := Vh ∩H1

0 (Ω),

and Vh is a NURBS space described in (19). In our framework we prefer to define this
space in the following general way, (see [1]):

V h = {vh ∈ H1
0 (Ω) : vh = v̂h ◦ x−1 ∈ V̂h}. (21)

V̂ h is a discrete space defined in the parametric domain Ω̂ such that

V̂ h =
{
vh : (0, 1)d −→ Rd | v̂h = vh ◦ x, vh ∈ V h}.

Note that the discrete problem (20) has a unique solution. Indeed, it is square system of
linear equations in finite dimension, and the integral in the right-hand side is well-defined
because the functions of V h belong to L∞(Ω).

We denote ah(uh, vh) :=

∫
Ω
A∇uh∇vhdx, the bilinear form of (20).

Since Splines are not in general interpolatory, a common way to define projection is by
giving a dual basis. Given a function v̂ ∈ L2(Ω̂) defined in the parametric domain Ω̂, we
use the projective operator over the B-Splines space Sh, say ΠSh

, introduced in [1] and
defined as:

ΠSh
: L2(Ω̂) → Sh, ΠSh

v̂ :=
∑
i∈I

λi(v̂)Ni, (22)

where the linear functionals λj ∈ L2(Ω̂)′ determine the dual basis for the set of B-Splines
[11], i.e. they are such that λj(Ni) := δj,i for i, j ∈ I. The corresponding projective
operator over the NURBS space Nh in the parametric domain (16), say ΠNh

, is defined



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 409

by means of ΠSh
and the definition of the NURBS basis functions of (15) through the

weighting function W of (14). In particular, ΠNh
reads:

ΠNh
: L2(Ω̂) → Nh, ΠNh

v̂ :=
ΠSh

(Wv̂)

W
, (23)

for all v̂ ∈ L2(Ω̂). In this manner, the projective operator over Vh, the NURBS space in
the physical domain Ω defined in (19) as the push-forward of the space Nh, is given by:

ΠVh : L2(Ω) → Vh, ΠVhv :=
(
ΠNh

(v̂)
)
◦ x−1. (24)

The following result is proved in [2] and shows that ΠVh is actually a projector on Vh.

Proposition 2.1. Its holds that ΠVhvh = vh for all vh ∈ Vh. That is, ΠVh is a projector.

Now, we define the real number

Dij =

∫
Ω
A∇λi∇λjdx; (25)

this defines an I × I matrix D.
Here, D satisfies

∀i ∈ I, Dii −
∑

j∈I,j ̸=i

|Dij | ≥ 0. (26)

D is assumed to be diagonally dominant matrix.
This assumption is close to the usual assumption which ensures the discrete maximum
principle.

Lemma 1. The matrix D(sh) is an M-matrix and satisfies property (26), ∀ sh ∈ V h.

The coerciveness of the bilinear form ah is the a consequence of this Lemma:

ah(vh, vh) ≥ α∥∇vh∥22, ∀vh ∈ V h. (27)

3. Convergence Analysis and Error estimates

In this Section, we give a priori estimates on the solution uh of (20). These results
allow to prove our main result.

Theorem 2. (see [4]) Assume that A satisfies (3) and (27). Then, for every h > 0, let uh

the unique solution of problem (20), then {uh}h>0 is bounded in W 1,q
0 (Ω) (1 ≤ q <

d

d− 1
)

and there exists a constant C > 0 independent of h, such that

∥uh∥W 1,q
0 (Ω)

≤ C∥f∥L1(Ω). (28)



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 410

Proposition 3.1. Under assumption (26), on has for every vh ∈ V h and every k > 0∫
Ω
A∇

(
vh −ΠVh(Tk(vh))

)
∇ΠVh(Tk(vh))dx ≥ 0. (29)

Proof. We use the technique applied in [3].
Since

vh =
∑
i∈I

vh(xi)λi and ΠVh(Tk(vh)) =
∑
i∈I

Tk(vh)(xi)λi,

using the definition (26) of Dij , we have∫
Ω
A∇

(
vh −ΠVh(Tk(vh))

)
∇ΠVh(Tk(vh))dx =

=
∑
i,j∈I

Dij

(
vh(xi)− Tk(vh(xi))

)
Tk(vh(xj)) =

∑
i∈I

Si,

where
xi = (ξi+1 + ...+ ξi+p)/p

and
Si = Dii

(
vh(xi)− Tk(vh(xi))

)
Tk(vh(xi))+

+
∑

j∈I,j ̸=i

Dij

(
vh(xi)− Tk(vh(xi))

)
Tk(vh(xj)).

Fix i ∈ I. If |vh(xi)| ≤ k, then vh(xi)− Tk(vh(xi)) = 0 and Si = 0.
If |vh(xi)| > k, then(

vh(xi)− Tk(vh(xi))
)
Tk(vh(xi)) = |vh(xi)− Tk(vh(xi))|k.

Since |Tk(vh(xj))| ≤ k for every j, one has

Si ≥ Dii|vh(xi)− Tk(vh(xi))|k −
∑

j∈I,j ̸=i

|Dij ||vh(xi)− Tk(vh(xi))|k

= |vh(xi)− Tk(vh(xi))|k
(
Dii −

∑
j∈I,j ̸=i

|Dij |
)
≥ 0,

owing the hypothesis (26). This proves that

∀i ∈ I, Si ≥ 0,

and therefore we obtain (29).

Now, we establish a priori estimate on the solution uh of (20).



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 411

Proposition 3.2. Under the assumptions (2.1), (3), (4), (10), (18) and (26). Then the
unique solution uh of (20) satisfies for every h > 0 and every k > 0∫

Ω
A∇ΠVh(Tk(uh))∇ΠVh(Tk(uh))dx ≤

∫
Ω
fΠVh(Tk(uh))dx. (30)

In particular, uh satisfies

α

∫
Ω
|∇ΠVh(Tk(uh))|2dx ≤ k∥f∥L1(Ω). (31)

Proof. Since Tk(uh) is continuous, the function ΠVh(Tk(uh)) belongs to Vh. Using this
function as test function in (20) we have∫

Ω
A∇uh∇ΠVh(Tk(uh))dx =

∫
Ω
fΠVh(Tk(uh))dx. (32)

Proposition (3.1) shows that∫
Ω
A∇

(
vh −ΠVh(Tk(vh))

)
∇ΠVh(Tk(vh))dx ≥ 0,

which immediately implies (30).
As a consequence, we consider (30) and the coercivity (3) of A to obtain (31).

Our main result is the following.

Theorem 3. Under the assumptions of Proposition (3.2), the unique solution uh of (20)

satisfies for every k > 0 and for every q with 1 ≤ q <
d

d− 1

uh −→ u strongly in W 1,q
0 (Ω), (33)

when the mesh size h tends to zero, where u is the unique renormalized solution of (1).

Proof. Let us consider
(
f ε

)
ε
, a sequence of functions such that

f ε ∈ L2(Ω), f ε −→ f strongly in L1(Ω).

We can take for example f ε = T 1
ε
(f).

Let uεh be the unique solution of problem (20) with regularized data f ε ∈ L2(Ω). Then
uh − uεh satisfies

uh − uεh ∈ V h,

∀vh ∈ V h,

∫
Ω
A∇(uh − uεh)∇vhdx =

∫
Ω
(f − f ε)vhdx.



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 412

We consider this problem and we apply estimate (31). We have for every k > 0, every
h > 0 and every ε > 0

α

∫
Ω

∣∣∣∇ΠVh

(
Tk(uh − uεh)

)∣∣∣2dx ≤ k∥f − f ε∥L1(Ω),

Next, applying Theorem 2.1 of [3] and using Theorem 2, we deduce that, for every q with

1 ≤ q <
d

d− 1
, every h > 0 and every ε > 0

∥uh − uεh∥W 1,q
0 (Ω)

≤ C2
1

α
∥f − f ε∥L1(Ω), (34)

where C2 is a constant which depends of d and q.
On the other hand, since f ε ∈ L2(Ω) and the fact that the physical mesh is quasi-uniform
under hypothesis (10) and (18), we have that, for any ε > 0

lim
h−→0

∥uεh − uε∥H1
0 (Ω) = 0, (35)

where uε is the unique solution of uε ∈ H1
0 (Ω),

−div(A∇uε) = f ε in D′(Ω).
(36)

Finally, the function uε, which is the unique weak solution of (36), is also the unique
renormalized solution in the sense of Definition 1.1 of the problem −div(A∇uε) = f ε in Ω,

uε = 0 on ∂Ω.
(37)

We consider u and uε the unique renormalized solutions of (1) and (37) respectively. We
have, indeed the continuous dependence of the renormalized solution with respect to the
data implies that

∥uε − u∥
W 1,q

0 (Ω)
≤ C3

1

α
∥f ε − f∥L1(Ω). (38)

for every q with 1 ≤ q <
d

d− 1
.

Inequality (38) is given by Theorem 1.2 in [3].
Writing now

∥uh − u∥
W 1,q

0 (Ω)
≤ ∥uh − uεh∥W 1,q

0 (Ω)
+ ∥uεh − uε∥

W 1,q
0 (Ω)

+ ∥uε − u∥
W 1,q

0 (Ω)

and using (34), (35) and (38), we have proved that for every ε > 0 and every q with

1 ≤ q <
d

d− 1

lim sup
h−→0

∥uh − u∥
W 1,q

0 (Ω)
≤

(
C1, C2, C3

) 1

α
∥f ε − f∥L1(Ω).



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 413

Taking the limit when ε tends to zero proves (33).

For every r with 1 < r < +∞, we denote by Lr,∞(Ω) the Marcinkiewicz space whose
norm is defined by

∥f∥Lr,∞(Ω) = sup
ν>0

(
ν
∣∣∣{x ∈ Ω : |f(x)| ≥ ν}

∣∣∣1/r). (39)

Next, error estimates for data f ∈ Lr,∞(Ω) may be derived using the techniques introduced
in [3].

Theorem 4. Under the assumptions of Theorem 2.2 and f ∈ Lr,∞(Ω) for some r with
1 < r < 2, there exists a constant C independent of the mesh size h such that we have the
error estimate

∥uh − u∥
W 1,q

0 (Ω)
≤ Ch2(1−

1
r
)∥f∥Lr,∞(Ω). (40)

Proof. We assume that f belongs to the Marcinkiewicz space Lr,∞(Ω) for some r with
1 < r < 2 (this holds in particular if f belongs to Lr(Ω) ). For every ε > 0, we set

f ε = T 1
ε
(f),

which belongs to L∞(Ω) ⊂ L2(Ω), and we denote by uεh the solution of (20) with right-hand

side f ε. Defining also uε at the solution of (36), we write for every q with 1 ≤ q <
d

d− 1

∥uh − u∥
W 1,q

0 (Ω)
≤ ∥uh − uεh∥W 1,q

0 (Ω)
+ ∥uεh − uε∥

W 1,q
0 (Ω)

+ ∥uε − u∥
W 1,q

0 (Ω)
. (41)

We have for a new constant C (which depends on q, Ω,)

∥uεh − uε∥
W 1,q

0 (Ω)
≤ Ch∥f ε∥L2(Ω).

Using then (34) and (38), we deduce that for a new constant C, which is independent of
ε, h and f (but depends on d, q,Ω), one has

∥uh − u∥
W 1,q

0 (Ω)
≤ C

(
∥f − f ε∥L1(Ω) + h∥f ε∥L2(Ω)

)
. (42)

We now estimate the right-hand side of this inequality by considering

∥g∥pLp(Ω) = p

∫ +∞

0
tp−1

∣∣∣{x ∈ Ω : |g(x)| ≥ t
}∣∣∣dt,

which gives 

∥f − f ε∥L1(Ω) =

∫ +∞

0

∣∣∣{x ∈ Ω : |f(x)− T 1
ε
(f)(x)| ≥ t

}∣∣∣dt
=

∫ +∞

0

∣∣∣{x ∈ Ω : |f(x)− 1

ε
| ≥ t

}∣∣∣dt
=

∫ +∞

1
ε

∣∣∣{x ∈ Ω : |f(x)| ≥ t
}∣∣∣dt,

(43)



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 414

and 
∥f ε∥2L2(Ω) = 2

∫ +∞

0
t
∣∣∣{x ∈ Ω : |T 1

ε
(f)(x)| ≥ t

}∣∣∣dt
= 2

∫ 1
ε

0
t
∣∣∣{x ∈ Ω : |f(x)| ≥ t

}∣∣∣dt. (44)

By the norm in the Marcinkiewicz space Lr,∞(Ω) define in (39) , we have∣∣∣{x ∈ Ω : |f(x)| ≥ t
}∣∣∣ ≤ min

{
|Ω|,

∥f∥rLr,∞(Ω)

tr

}
,

and thus 
∥f − f ε∥L1(Ω) ≤ 1

r − 1
εr−1∥f∥rLr,∞(Ω),

∥f ε∥L2(Ω) ≤
√

2

2− r

1

ε1−
r
2

∥f∥
r
2

Lr,∞(Ω).

(45)

Then, (42) gives

∥uh − u∥
W 1,q

0 (Ω))
≤ C

( 1

r − 1
εr−1∥f∥rLr,∞(Ω) +

√
2

2− r

h

ε1−
r
2

∥f∥
r
2

Lr,∞(Ω)

)
.

Taking in this inequality ε =
h

2
r

∥f∥
r
2

Lr,∞(Ω)

yields, for every q with 1 ≤ q <
d

d− 1
and for

every h > 0, we obtain

∥uh − u∥
W 1,q

0 (Ω)
≤ C

(
d, q, r, |Ω|,

)
h2(1−

1
r
)∥f∥rLr,∞(Ω).

4. Numerical implementation

In this Section, we give the numerical tests to attest our main error estimate result,
namely Theorem (4). We consider in this paper for the numerical test a simple geometry :
a quarter of a ring with inner and outer radius equal to 1 or 2, respectively, and described
through a quadratic NURBS parametrization, as the one in Figure 1.
We solve the initial problem (1) with A(x) the identity matrix. This matrix satisfy the
assumptions of Theorem (3), namely (2), (3), (4), (10), (18) and (26), are satisfied. The
right-hand side f is imposed to obtain the renormalized solution u = ex1 sin(x2).
We solve the problem ina set of successively refined meshes, the coarest three meshes are
plotted in Figure 1, for degree p varying from 2 to 4, and in NURBS spaces of maximum
(Cp−1) and minimum (C0) continuity.

In Figure 2, we present the error in the W 1,q
0 -norm with respect to the mesh size h,

and with respect to the number of degree of freedom. The result in terms of the mesh size
confirm the estimate of Theorem (4) when we take for example



Y. C. Bassonon, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (1) (2023), 404-417 415

Figure 1: Mesh parametrization.

f(x) =
1

|x|2/r
∈ Lr,∞(Ω).

In terms of the degrees of freedom, the results always converges like O(N
−p/2
dof ) where Ndof

is the number of degrees of freedom

(a) Error in the terms of the mesh size. (b) Error in terms of the degrees of freedom.

Figure 2: Error estimates in the W 1,1
0 - norm in the quarter-ring: error in terms of (A) the

mesh size, and (B) the degrees of freedom.



REFERENCES 416

5. Conclusion

In this paper, we discussed in dimension d ≥ 2, the Isogeometric Analysis approxima-
tion of second order elliptic equations in divergence form with right-hand side in L1. We
have proven that the unique solution of the discrete problem converges, in NURBS space,

to the unique renormalized solution in W 1,q
0 (Ω), 1 ≤ q <

d

d− 1
. We have also studied

the convergence analysis and we have obtained the error estimates for data in Lr;∞(Ω) for
1 < r < 2. To finish, we gave numerical results using Python.

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