# Guessab AllalUniversité de Pau et des Pays de l'Adour | UPPA · Department of Mathematics

Guessab Allal

University Professor

## About

109

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Introduction

Election campaign communication research

## Publications

Publications (109)

The aim of this paper is to unify the ideas and to extend to a more general setting the work done in [1] for a polynomial enrichment of the standard three-node triangular element (triangular
linear element) using line integrals and quadratic polynomials. More precisely, we introduce a new class of nonconforming finite elements by enriching the clas...

We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus φ. Inspired by three classical characterizations of ordinary convexity, we aim at characteri...

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular Shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of...

Delaunay triangulations and Voronoi diagrams are important tools for problem resolution in numerous scientific fields. This readable book provides a comprehensive, rigorous and self-contained of these two geometrical structures and their connections with sharp approximations. The theory developed includes some old and some new results, concepts, an...

Delaunay triangulations and Voronoi diagrams are important tools in many fields like Astronomy, Physics, Chemistry, Biology, Ecology, Economics, Mathematics and Computer Science. They are oftentimes used in problems related to the generation meshes. The main purpose of this book is to study the interactions between these two geometrical structures...

In the present investigation, we introduce and study linear operators, which underestimate every strongly convex function. We call them, for brevity, sp-linear (approximation) operators. We will provide their sharp approximation errors. We show that the latter is bounded by the error approximation of the quadratic function. We use the centroidel Vo...

In this paper, we introduce an extension of our previous paper, A globallyconvergent version to the Method of Moving Asymptotes, in a multivariatesetting. The proposed multivariate version is a globally convergent result fora new method, which consists iteratively of the solution of a modified versionof the method of moving asymptotes. It is shown...

We consider the d-dimensional Hermite–Hadamard inequality
$$\displaystyle {} \frac {1}{\left |S\right |} \int _{S}f({\boldsymbol x}) \, d{\boldsymbol x} \leq Q^{\text{ tra}}(f):= \frac {1}{\left |\partial S\right |} \int \limits _{\partial S}f({\boldsymbol x})d\gamma . $$
(1)
Here f is a convex function defined on a simplex \(S\subset \mathbb {R}^d...

In this paper, we are interested in the problem of approximation of a definite integral over a ball of a given function f in d-dimensional space when, rather than function evaluations, a number of integrals over certain (d−1)-dimensional hyperspheres are only available. In this context several families of ‘extended’ multidimensional integration for...

This paper is devoted to study and construct a family of multidimensional numerical integration formulas (cubature formulas), which approximate all strongly convex functions from above. We call them strongly negative definite cubature formulas (or for brevity snd-formulas). We attempt to quantify their sharp approximation errors when using continuo...

In this paper, we introduce an extension of our previous paper, A globally convergent version to the Method of Moving Asymptotes, in a multivariate setting. The proposed multivariate version is a globally convergent result for a new method, which consists iteratively of the solution of a modified version of the method of moving asymptotes. It is sh...

This paper introduces and studies a new class of multidimensional numerical integration, which we call ?strongly positive definite cubature formulas?. We establish, among others, a characterization theorem providing necessary and sufficient conditions for the approximation error (based on such cubature formulas) to be bounded by the approximation e...

A new modified moving asymptotes method is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and explicitly solved. In doing so we propose a strategy to incorporate a modified second-order information for the moving asymptotes location. Under natural assumptions, we prove the geometrical conve...

In this paper we will determine the best constant for a class of (weighted and non-weighted) new Poincaré-type inequalities. In particular, we obtain sharp inequalities under the concavity, convexity of the weight function. We also establish a family of sharp Poincaré inequalities involving the second derivative.

For any positive integer n, let Tkk=1n be a given set of linear functionals on W1n(0,1), which are unisolvent for polynomials of degree n − 1. We determine the best possible constant c(T1, …, Tn) in the following general higher-order Poincaré-type inequalities ∫01f(x)dx≤cn(T1,…,Tn)∫01|f(n)(x)|dx, where f∈W1n(0,1) satisfying the conditions Tkf=0,k =...

In this paper, we consider a general decomposition of any convex polytope P \subset \BbbR ninto a set of subpolytopes \Omega i and develop methods for approximating a definite integral of a givenfunction f over P when, rather than its values at some points, a number of integrals of f over thefaces of \Omega i are only available. We present several...

We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus ?. Inspired by three classical characterizations of ordinary convexity, we aim at characteri...

A new modified moving asymptotes method is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and explicitly solved. In doing so we propose a strategy to incorporate a modified second-order information for the moving asymptotes location. Under natural assumptions, we prove the geometrical conve...

In this work, Lp-error estimates of general two and three point quadrature rules for Riemann-Stieltjes integrals are give n. The presented proofs depend on new triangle type inequalities of Riemann-Stieltjes integrals

A new modified moving asymptotes method is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and explicitly solved, and in doing so we propose a strategy to incorporate a modified second-order information for the moving asymptotes location. This considerably reduces the computational cost of o...

This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity pro...

In this work, L p -error estimates of general two and three point quadrature rules for Riemann-Stieltjes integrals are given. The presented proofs depend on new triangle type inequalities of Riemann-Stieltjes integrals.

In this paper, we consider the problem of approximating a definite integral of
a given function f when, rather than its values at some points, a number of
integrals of f over some hyperplane sections of simplices in a triangulation of a
polytope P in Rd are only available. We present several families of integration
formulas, all of which are a weig...

According to Hammer’s inequality (Hammer in Math Mag 31:193–195, 1958), which is a refined version of the famous Hermite–Hadamard inequality, the midpoint rule is always more accurate than the trapezoidal rule for any convex function defined on some real numbers interval [a, b]. In this paper we consider some properties of a multivariate extension...

Colloquium in Honor of Prof. Dr. Gerhard Schmeisser Department of Mathematics, University of Erlangen-Nuremberg

This paper focuses on the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over certain hyperplane sections of a d-dimensional hyper-rectangle Cd are only available. We develop several families of integration formulas, all of which are a weighted sum of integr...

In this chapter our goal is to develop a unified and general framework for enriching finite element approximations via the use of additional enrichment functions. A crucial point in such an approach is to determine conditions on enrichment functions which guarantee that they generate a well-defined finite element. We start by giving under some cond...

We provide a simple condition, which is both necessary and sufficient, that guarantees the existence of an
enriched Crouzeix-Raviart element. Our main result shows that the latter
can be easily expressed in terms of the approximation error in a multivariate generalized trapezoidal type cubature formula.
Furthermore, we
derive simple explicit formul...

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enric...

This paper establishes a new class of nonconforming finite elements by including additional enriched functions (not necessary polynomials) to the standard Q1(K) element on convex polytope. Here we focus on their fundamental construction principles and some of their approximation properties. In addition, we show how this approach can be used to enri...

This papere stablishes a general approach for constructing a new class of nonconforming finite elements on arbitrary convex polytope.Our contributions generalize or completes everal well-known nonconforming finite elements such as: theCrouzeix–Raviart triangle element,the Han parallelogram element, the nonconforming rotated parallelogram element of...

This paper deals with the problem of finding lower and upperbounds
in a set of convex functions to a given positive linear functional; that
is, bounds which estimate always below (or above) a functional over a family of
convex functions. A new set of upper and lower bounds are provided and their
extremal properties are established. Moreover, we sho...

This paper and refines a result due to Fejes (1939).

Recently there has been renewed interest in the problem of �nding under
and over estimations on the set of convex functions to a given non-negative
linear functional; that is, approximations which estimate always below (or
above) the functional over a family of convex functions. The most important
example of such an approximation problem is given b...

The aim of this paper is to find a convenient and practical method to approximate a given real-valued function of multiple variables by linear operators, which approximate all strongly convex functions from above (or from below). Our main contribution is to use this additional knowledge to derive sharp error estimates for continuously differentiabl...

Univariate symmetrization technique has many good properties. In this paper, we adopt the high-dimensional viewpoint, and propose a new symmetrization procedure in arbitrary (convex) polytopes of $\R^d$ with central symmetry.
Moreover, the obtained results are used to extend to the arbitrary centrally symmetric polytopes
the well-known Hermite-Hada...

Let Xn≔{xi}i=0n be a given set of (n+1)(n+1) pairwise distinct points in RdRd (called nodes or sample points), let P=conv(Xn)P=conv(Xn), let f be a convex function with Lipschitz continuous gradient on P and λ≔{λi}i=0n be a set of barycentric coordinates with respect to the point set XnXn. We analyze the error estimate between f and its barycentric...

Based on permutation enumeration of the symmetric group and `generalized'
barycentric coordinates on arbitrary convex polytope, we develop a
technique to obtain symmetrization procedures for functions that provide a
uni�ed framework to derive new Hermite-Hadamard type inequalities. We
also present applications of our results to the Wright-convex fu...

In this paper we propose new local convex approximations for solving unconstrained
non-linear optimization problems, based on a moving asymptotes algorithm. This method provides
the second order information for the moving asymptotes location. As a consequence, at each step of
the iterative process, a strictly convex approximation sub-problem is gen...

In this paper we propose new local convex approximations for solving unconstrained
non-linear optimization problems, based on a moving asymptotes algorithm. This method provides
the second order information for the moving asymptotes location. As a consequence, at each step of
the iterative process, a strictly convex approximation sub-problem is gen...

In this paper we propose new local convex approximations for solving unconstrained non-linear
optimization problems based on a moving asymptotes algorithm. This method incorporates second-order information for the moving asymptotes location. As a consequence, at each step of the iterative process, a strictly convex approximation subproblem is gener...

Recently there has been renewed interest in the problem of finding under
and over estimations on the set of convex functions to a given non-negative
linear functional; that is, approximations which estimate always below (or
above) the functional over a family of convex functions. The most important
example of such an approximation problem is given...

Recently there has been renewed interest in the problem of ﬁnding under
and over estimations on the set of convex functions to a given non-negative
linear functional; that is, approximations which estimate always below (or
above) the functional over a family of convex functions. The most important
example of such an approximation problem is given b...

We consider the d-dimensional Jensen inequality T [ϕ(f1,. .. , f d)] ≥ ϕ(T [f1],. .. , T [f d ]) (*) as it was established by McShane in 1937. Here T is a functional, ϕ is a convex function defined on a closed convex set K ⊂ R d , and f1,. .. , f d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for t...

In this paper we obtain some direct and converse new multidimensional Jensen's type inequalities on convex polytopes. Among the inequalities presented, we offer, as a particular case of our general results, a direct and converse multivariate extension of Mercer inequality. The main results are obtained with the help of the generalized barycentric c...

In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants, which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the...

In this paper, we study the error in the approximation of a convex function obtained
via a one-parameter family of approximation schemes, which we refer to as barycentric
approximation schemes. For a given finite set of pairwise distinct points Xn := {xi}
n
i=0
in
R
d
, the barycentric approximation of a convex function f is of the form:
B[f ](x) =...

We consider the d-dimensional Jensen inequality
$$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set \({K\subset \mathbb{R}^d}\) , and f
1, . . . , f
d
are from some linear space of functions. O...

This paper deals with the problem of finding lower and upper-bounds in a set of convex functions to a given positive linear functional; that is, bounds which estimate always below (or above) a functional over a family of convex functions. A new set of upper and lower bounds are provided and their extremal properties are established. Moreover, we sh...

a b s t r a c t In this paper, we propose several approximations of a multivariate function by quasi-interpolants on non-uniform data and we study their properties. In particular, we char-acterize those that preserve constants via the partition of unity approach. As one of the main results, we show how by a very simple modification of a given quasi...

Mercer's 'a variant of Jensen's inequality' for functions of one vari-able in [22] is shown to be a special case of a refinement of Jensen's inequality, available for any multivariate convex function defined on a convex polytope Ω in the d-dimensional Euclidean space. In addition, we also examine the converse inequality for Mercer's result under ap...

We consider four-point subdivision schemes of the form
$$ (Sf)_{2i} = f_i,\qquad (Sf)_{2i+1} = \frac{f_i+f_{i+1}}{2} - \frac{1}{{8}}M\!\left(\strut \Delta^2f_{i-1}, \Delta^2f_i\right) $$
with any M that is originally defined as a positive-valued function for positive arguments and is extended to the whole of ℝ2 by setting
$M(x,y):=- M(\left|x\rig...

In this paper, we present a more general and extended form in a multivariate setting of an inequality of Brenner and Alzer. Mathematics Subject Classification (2010). Primary 41A36, 41A63, 41A80, 47A58; Secondary 26B25, 47B65.

Given a B-spline M on R^s, s>=1 we consider a classical discrete quasi-interpolant Q"d written in the formQ"df=@?i@?Z^sf(i)L(@?-i),where L(x)@?@?"j"@?"Jc"jM(x-j) for some finite subset J@?Z^s and c"j@?R. This fundamental function is determined to produce a quasi-interpolation operator exact on the space of polynomials of maximal total degree includ...

a b s t r a c t In [A. Guessab, O. Nouisser, G. Schmeisser, Multivariate approximation by a combination of modified Taylor polynomials, J. Comput. Appl. Math. 196 (2006) 162–179], a general method is proposed to increase the approximation order of approximation operators. In this work, by using these enhancement techniques, we introduce and study n...

Let Ω ⊂ ℝd be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the...

Let Ω⊂ℝd
be a compact convex polytope of positive measure. We study cubature formulae on Ω which approximate the integral of every convex function f∈C(Ω) from above. They are called negative definite formulae or nd-formulae for short. In particular, we characterize nd-formulae by certain partitions of unity or, alternatively, by a class of positive...

Let Ω be a compact convex domain in
\mathbbRd{\mathbb{R}}^{d} and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients amrj the operator
Hmr[f](x): = L [åj=0r \fracamrjj! Djx-f ] (x) (x Î W)H_{mr}[f]({\bf x}):= L \left[...

Let $\Omega$ be a compact convex subset of $\R^d$ and let $(L_n)_{n\in\N}$ be
a sequence of positive linear operators that map $C(\Omega)$ into itself.
We establish two Korovkin-type theorems in which the limit of the sequence
of operators is not necessarily the identity.

Let Ω ⊂ ℝd be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short)
if it approximates the integral ∫Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable
functions. In the univariate case (d = 1), t...

We study an approximation of a multivariate function f by an operator of the form N i=1 T r [f, x i ](x) i (x), where 1 , . . . , N are certain basis functions and T r [f, x i ](x) are modified Taylor polynomials of degree r expanded at x i . The modification is such that the operator has highest degree of algebraic precision. In the univariate cas...

Let q3 be a convex polytope in the d-dimensional Euclidean space. We consider an interpolation of a function f at the vertices of ~ and compare it with the interpolation of f and its derivative at a fixed point y E 3. The two methods may be seen as multivariate analogues of an interpolation by secants and tangents, respectively. For twice continuou...

RESUMEN RESUMEN
Let $\ mathfrak { P}$ be a convex polytope in the d -dimensional Euclidean space. We consider an interpolation of a function f at the vertices of $\ mathfrak {P}$ and compare it with the interpolation of f and its derivative at a fixed point $y\in\ mathfrak {P}.$ The two methods may be seen as multivariate analogues of an interpola...

Abstra&--Extensions of quadrature formulae are of importance, for example, in the construction of automatic integrators, but many sequences fail to exist in usable form. Using the theory of quasiorthogonality and reinterpreting it in terms of the standard orthogonal polynomials, we find an approximation of the integral by a convex combination of th...

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite–Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex func-tion. We establish multivariate analogues of the Hermite–Hadamard inequal-ities and obtain access to multivariate integration formula...

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pec ˘ari, and others are obtained as special cases. Our results can also be interpreted as analogues to a the...

Starting from two sequences Ga;c;n} and G d; b; n } of ordinary Gauss quadrature formulae with an orthogonality measure d on the open intervals (a; c) and (d; b), respectively. We construct a new sequence G a; b; e(n) } of extended Gaussian quadrature formulae for d on (a; b), which is based on some preassigned points, the nodes o Ga;c;n G d; b; n...

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors...

For every even k, 0<k<2, we show the existence of a quadrature formula on [-1,1] which uses f (k) (0) and the value of f at n distinct points and integrates exactly all polynomials of degree 2n+1. Then we apply this quadrature to answer a question raised by Turán.

In this paper we prove several inequalities for polynomials and trigonometric polynomials. They are all obtained as applications of certain quadrature formulae, some of which are proved here for the first time. Such an application of a Gaussian quadrature formula was pointed out by Bojanov in 1986 (see East. J. Approx. 1 (1995), 37746; J. Approx. T...

Let Pn be the class of algebraic polynomials of degree at most n and ||P|| = (∫|P(t)|2 dσ(t))1/2, where da(t) is a measure corresponding to the classical orthogonal polynomials. We study extremal problems of Markov's type [equation presented] where A is given by (1-1), and the differential operator Dm is defined by (1.3). The best constants are fou...

In this paper we give a new characterization of the classical orthogonal polynomials (Hermite, generalized Laguerre, Jacobi) by extremal properties in some weighted polynomial inequalities in L2-norm.

An assumed strain mixed finite element method is presented for fully nonlinear problems in solid mechanics, restricted to rubber-like materials. The method relies on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three-field variational formulation. This method...

Let n be the class of algebraic polynomials of degree at most n. Some weighted L2-analogues of the Bernstein′s inequality for polynomials P ∈ n are investigated and a connection with the classical orthogonal polynomials is given.

Let Pn be the class of algebraic polynomials P(x) = ∑n ν = 0aν xν of degree at most n and |P|dσ = (∫R|P(x)|2 dσ(x))1/2, where dσ(x) is a nonnegative measure on R. We determine the best constant in the inequality |aν| ≤ Cn,ν(dσ)|P|dσ, for ν = n and ν = n - 1, when P ∈ Pn and such that P(ξk) = 0, k = 1, ..., m. The case dσ(t) = dt on [ -1, 1 ] and P(...

n assumed strain mixed finite element method for fully nonlinear problems in solid mechanics,
restricted to rubber-like materials is presented. The method relies on a local multiplicative decomposition
of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three
field variational formulation. This method...

Let P n {\mathcal {P}_n} be the class of algebraic polynomials P ( x ) = ∑ v = 0 n a ν x ν P(x) = \sum \nolimits _{v = 0}^n {{a_\nu }{x^\nu }} of degree at most n n and | | P | | d σ = ( ∫ R | P ( x ) | 2 d σ ( x ) ) 1 / 2 ||P|{|_{d\sigma }} = {({\smallint _\mathbb {R}}|P(x){|^2}d\sigma (x))^{1/2}} , where d σ ( x ) d\sigma (x) is a nonnegative mea...

We will consider the problem of approximating a double integral on a convex compact set K as a minimal linear combination of integrals on the real line. We obtain Cubature formulae, which are exact on the space Q
2k+1(K) of all polynomials of degree 2k + 1 respectively to each variable x
i
, i = 1, 2.

Let W n be the set of all polynomials of exact degree n, whose coefficients are nonnegative. For the norm in L 2 [0,b], b>0, with generalized Laguerre weight function w(t)=t α e -t (α>-1), the extremal problem A n,2 (1) (α)=sup P∈W n ∥P ' ∥ 2 ∥P∥ 2 +[e -t ((t α P(t) 2 )+(t α P(t) 2 ) ' )] t=b is solved. Also, some generalizations for higher derivat...

## Projects

Projects (6)

I am interested in establishing optimal general Hermite-Hadamard-type inequalities on convex sets.
The following publications fully demonstrate the important applications of these type of inequalities in multidimensional numerical integration and nonconforming finite element methods.
• Guessab, A.; Schmeisser, G. Necessary and sufficient conditions for the validity of Jensen's inequality. Arch. Math. (Basel) 100 (2013), no. 6, 561–570.
• Guessab, Allal; Nouisser, Otheman; Pečarić, Josip A multivariate extension of an inequality of Brenner–Alzer. Arch. Math. (Basel) 98 (2012), no. 3, 277–287.
• Guessab, Allal, Direct and converse results for generalized multivariate Jensen-type inequalities.
J. Nonlinear Convex Anal. 13 (2012), no. 4, 777–797.
• Guessab, Allal; Schmeisser, Gerhard Sharp integral inequalities of the Hermite-Hadamard type. J. Approx. Theory 115 (2002), no. 2, 260–288.
• Guessab, Allal; Schmeisser, Gerhard Sharp error estimates for interpolatory approximation on convex polytopes. SIAM J. Numer. Anal. 43 (2005), no. 3, 909–923.
• Guessab, Allal; Schmeisser, Gerhard Convexity results and sharp error estimates in approximate multivariate integration. Math. Comp. 73 (2004), no. 247, 1365–1384.
• Guessab, Allal Approximations of differentiable convex functions on arbitrary convex polytopes. Appl. Math. Comput. 240 (2014), 326–338.
• Guessab, Allal Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes. Comput. Math. Appl. 66 (2013), no. 6, 1120–1136.
• B. Achchab, A. Agouzal, A. Guessab, Y. Zaim, An extended family of non-conforming quasi-Wilson elements for solving
elasticity problem, Applied Mathematics and Computation, Volume 344-345:1-19, 2018.
• M. Bachar, A. Guessab, O. Mohammed, Y. Zaim, New Cubature formulas and Hermite-Hadamard type inequalities
using integrals over some hyperplanes in the d-dimensional hyper-rectangle, App. Math. and Comp., 315:347-362, 2017.
• A. Guessab, Y. Zaim, A unified and general framework for enriching finite element approximations, chapter of the
Progress in Approximation Theory and Applicable Complex Analysis, Volume 117 of the series, Springer Optimization
and Its Applications, 2017.
• B. Achchab, A. Guessab, Y. Zaim, A new class of nonconforming finite element for enrichment of Q1 element on convex
polytope, Applied Mathematics and Computation, Volume 271, 657-668, 2015.