Two one-parameter families of nonconforming enrichments of the Crouzeix-Raviart finite element
Federico Nudo









  address-CS,address-Pau]Federico Nudocorrauthor
  [corrauthor]Corresponding author
  federico.nudo@unical.it


  [address-CS]Department of Mathematics and Computer Science, University of Calabria, Rende (CS), Italy
  [address-Pau]Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142, Université de Pau et des Pays de l'Adour (UPPA), 64000 Pau, France





In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.



Crouzeix–Raviart finite elementEnriched finite element method Nonconforming finite element Beta function






§ INTRODUCTION

The analysis of various physical phenomena often involves the widespread application of the finite element method, a powerful and versatile tool for approximating solutions of partial differential equations (PDEs) <cit.>. One of the reasons for its undeniable popularity lies in its ability to handle various types of geometries. In this method, the problem domain is divided into polytopes known as finite elements. Each finite element represents a local solution of the differential problem, and their collective contributions construct a global piecewise solution. If the global piecewise solution remains continuous at the boundary of the subdomains, the finite element is referred to as conforming; otherwise, it is referred to as nonconforming.

A finite element is locally defined as the triple ℳ_d=(S_d, 𝔽_S_d, Σ_S_d), where <cit.>

       
  * S_d is a polytope in ℝ^d, d≥ 1,
       
  * 𝔽_S_d  is a vector space of dimension n formed by real-valued functions defined on S_d, also known as trial functions,
       
  * Σ_S_d={L_j   :   j=1,…,n} is a set of linearly independent linear functionals from the vector space 𝔽_S_d, referred to as degrees of freedom,

ensuring that 𝔽_S_d is Σ_S_d-unisolvent, meaning that if f∈𝔽_S_d and 

    L_j(f)=0,      j=1,…,n,

then f=0. To enhance the local approximation accuracy in the finite element method, a commonly employed technique involves introducing enrichment elements <cit.>. Specifically, to enrich the finite element ℳ_d, we introduce a set of enrichment functions e_1,…,e_N and a set of enriched linear functionals

    Σ^enr_S_d={L_j  :   j=1,…,n+N},

ensuring that the approximation space 𝔽_S_d is Σ^enr_S_d-unisolvent. We then define the enriched finite element

    ℳ^enr_d=(S_d,𝔽^enr_S_d, Σ^enr_S_d),

where

    𝔽^enr_S_d=𝔽_S_d⊕{e_1,…,e_N}.

 The crucial aspect lies in the selection of the enrichment functions e_1,…,e_N to ensure that the triple (S_d, 𝔽^enr_S_d, Σ^enr_S_d) is a finite element. 


The Crouzeix–Raviart finite element, named in honor of mathematicians Vivette Crouzeix and Pierre-Arnaud Raviart, is widely used in numerical analysis for numerically solving second-order elliptic PDEs. Its nonconforming nature proves highly effective in handling challenges posed by intricate geometries and irregular meshes <cit.>. Within the realm of the Crouzeix–Raviart finite element method, the careful selection of approximation spaces plays a crucial role. These spaces are chosen to efficiently represent discontinuous solutions, making the method particularly well-suited for problems with singularities or discontinuities. This distinctive feature has established the Crouzeix–Raviart finite element as a valuable tool across various fields, including fluid dynamics, structural mechanics, and other field where obtaining accurate and flexible numerical solutions to PDEs is essential <cit.>.
In this paper, we present two one-parameter families of quadratic polynomial enrichments for the classical Crouzeix–Raviart finite element. These enrichments are obtained by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions.



The paper is organized as follows. In Section <ref>, we introduce a new one-parameter family of enrichment of the classical Crouzeix–Raviart finite element using quadratic polynomial functions. Additionally, we provide an explicit expression for the associated basis functions and introduce a new quadratic approximation operator based on this enriched finite element. In line with the previous section, in Section <ref> we propose another class of enrichment of the Crouzeix–Raviart finite element by employing quadratic polynomial functions. Numerical results are presented in Section <ref>. 





§ A FIRST ONE-PARAMETER FAMILY OF ENRICHMENT OF THE CROUZEIX–RAVIART FINITE ELEMENT

Let T ⊂ℝ^2 be a nondegenerate triangle with vertices v_1, v_2, v_3. Let λ_i denote the barycentric coordinate associated to the vertex v_i, and let Γ_i represent the edge of T opposite the vertex v_i, i=1,2,3. These functions are affine and satisfy the following properties

    λ_i(v_j) = δ_ij,
         i,j=1,2,3,

and

    λ_i(t x + (1-t) y) = t λ_i(x) + (1-t)λ_i(y),     x, y∈ T,

where δ_ij is the Kronecker delta function, that is

    δ_ij=
       1,   if  i=j, 
       0,   if  i≠ j.

By (<ref>) and (<ref>), we immediately get

    λ_i(x)=0,     x∈Γ_i,      i=1,2,3.

Hereafter, we adopt the standard convention that 

    v_4=v_1,   v_5=v_2

and that 

    λ_4=λ_1,   λ_5=λ_2.

We denote by ℐ_j^CR, j=1,2,3, the following linear functional

    ℐ_j^CR(f) = ∫_0^1f(tv_j+1+ (1-t)v_j+2)  dt,      j=1,2,3.

The Crouzeix–Raviart finite element is defined locally as the triple

    (T,ℙ_1(T), Σ_T^CR),

 where ℙ_1(T) is the space of all bivariate linear polynomials, that is
 
    ℙ_1(T)=span{λ_1,λ_2,λ_3}

 and 
 
    Σ_T^CR={ℐ_j^CR  :   j=1,2,3}.

The main goal of this section is to construct a parametric family of quadratic enrichments of the Crouzeix–Raviart finite element. To achieve this, we introduce a real parameter α > -1. Using this parameter, we define the following enriched linear functionals
  
    ℱ_j,α^enr(f) = ∫_0^1w_α(t)f(t v_j+ (1-t)
     m_j)  dt,      j=1,2,3,
 
 where 

    m_1= v_2+ v_3/2∈Γ_1,      m_2= v_3+ v_1/2∈Γ_2,      m_3= v_1+ v_2/2∈Γ_3,

 and

    w_α(t) = t^α (1 - t)^α

is the Jacobi weight function defined on [0,1].
The one-parameter family of triples is then defined as
 
    𝒞_α= (T,  ℙ_2(T),Σ_T,α^enr),

 where ℙ_2(T) is the space of all bivariate quadratic polynomials, that is
 
    ℙ_2(T)=span{λ_1,λ_2,λ_3, λ_1^2,λ_2^2,λ_3^2}
   
 and
 
    Σ_T,α^enr=  {ℐ_j^CR,ℱ^enr_j,α  :   j=1,2,3}

is the set of enriched degrees of freedom. 
In the following, we prove that for 

    α≠ -6/7,

 the triple 𝒞_α defined in (<ref>) is a finite element. To establish this, we require some preliminary results.
First and foremost, we unveil explicit formulations for the basis functions of ℙ_2(T)

    ℬ_AF3={φ_i, ϕ_i  :   i=1,2,3},

designed to satisfy conditions <cit.>

    ℒ^enr_j(φ_i)=δ_ij,   ℐ_j^CR(φ_i)=0,    i,j=1,2,3,


    ℒ^enr_j(ϕ_i)=0,   ℐ_j^CR(ϕ_i)=δ_ij,    i,j=1,2,3,

where

    ℒ_j^enr(f)= f(v_j),      j=1,2,3.

These functions are recognized as the basis functions of ℙ_2(T) associated to the enriched finite element <cit.>

    AF3= (T,ℙ_2(T),Σ_T^enr),

where

    Σ_T^enr= {ℐ_j^CR,ℒ^enr_j  :   j=1,2,3}.




        The basis functions φ_i,ϕ_i,  i=1,2,3, of ℙ_2(T) associated to the finite element AF3 have the following expressions

    φ_1 = λ_1 (1-  3λ_2  -3 λ_3),     φ_2=λ_2 (1-  3λ_1  -3 λ_3),     φ_3 = λ_3 (1-  3λ_1  -3 λ_2),


    ϕ_1 = 6 λ_2λ_3,    ϕ_2 = 6 λ_1λ_3,    ϕ_3 = 6λ_1λ_2.

      

    We remark that any p∈ℙ_2(T) can be expressed as
    
    p=∑_k=1^3  ℒ^enr_k(p)φ_k+∑_k=1^3  ℐ^CR_k(p)ϕ_k.




        We observe that, as a consequence of (<ref>) and (<ref>), we obtain
 
    λ_i( m_j)=1/2(1-δ_ij),      i,j=1,2,3.
 


The next lemma plays a crucial role in deriving the main result of this section. To aid in this effort, we introduce the following notations

    γ_α =  B(α+2,α+1),      
       h_α=-5α+6/4(2α+3),   
       K_α=-α/2α+3,

 where 

    B(z_1,z_2)=∫_0^1u^z_1-1(1-u)^z_2-1 du,      z_1,z_2>-1,

is the classical Euler beta function <cit.>. This function is symmetric, i.e. 

    B(z_1,z_2)=B(z_2,z_1),      z_1,z_2>-1,

and satisfies

    B(z_1+1,z_2)=z_1/z_1+z_2B(z_1,z_2),       z_1,z_2>-1.




  Let p ∈ℙ_2(T) such that ℐ_j^CR(p)=0, j=1,2,3. Then, the values of p at the vertices of T satisfy the following linear system 
    γ_α[ K_α h_α h_α; h_α K_α h_α; h_α h_α K_α ][ p( v_1); p( v_2); p( v_3) ]
    = [ ℱ_1,α^enr(p); ℱ_2,α^enr(p); ℱ_3,α^enr(p) ].

  


 Let p ∈ℙ_2(T) such that
 
    ℐ_j^CR(p)=0,      j=1,2,3.

Referring to (<ref>) and (<ref>), we can express p as follows 
 
    p=∑_k=1^3ℒ^enr_k(p)φ_k+∑_k=1^3 ℐ^CR_k(p) ϕ_k= ∑_k=1^3 p( v_k)φ_k.

Applying the functional ℱ_j,α^enr, j=1,2,3, to both sides of (<ref>), we obtain 

    ℱ_j,α^enr(p) = ∑_k=1^3 p(v_k)ℱ_j,α^enr(φ_k),      j=1,2,3.

Using (<ref>) in combination with (<ref>) and (<ref>), and leveraging the properties of the Euler beta function (<ref>) and (<ref>), straightforward calculations lead to the following results

    ℱ_j,α^enr(φ_j)    =    K_αγ_α,      j=1,2,3,
    
    ℱ_j,α^enr(φ_k)    =    h_αγ_α,      j,k=1,2,3,    j≠ k.

By substituting (<ref>) and (<ref>) into (<ref>), the thesis follows.




  Let α > -1 be a fixed real number different from -6/7 and let p∈ℙ_2(T) such that ℐ_j^CR(p)=0, j=1,2,3. Then p vanishes at the vertices of the triangle T if and only if

    ℱ_j,α^enr(p)= 0,    j=1,2,3.

 
  Let p∈ℙ_2(T) such that 
   
    ℐ_j^CR(p) = 0,      j=1,2,3.

By leveraging (<ref>) of Lemma <ref>, it follows that, if p vanishes at the vertices of the triangle T, then condition (<ref>) is satisfied. It remains to prove the reverse implication. Then we assume that p satisfies condition (<ref>). In this hypothesis, the linear system (<ref>) becomes
 
    γ_α[ K_α h_α h_α; h_α K_α h_α; h_α h_α K_α ][ p( v_1); p( v_2); p( v_3) ]
    = [ 0; 0; 0 ],

where γ_α,h_α, K_α are defined in (<ref>).
The required result follows since the determinant of the matrix associated to the above linear system is 
 
    Δ_α=- γ_α^3(6 + α)^2 (6 + 7 α)/32(3 + 2 α)^3,

 which is different from zero for any α>-1 and α≠ -6/7. 
 



  
  For any real number α > -1, and different from -6/7, the triple 𝒞_α is a finite element.


Let p∈ℙ_2(T) such that

    ℐ_j^CR(p)    =    0,      j=1,2,3, 
    ℱ_j,α^enr(p)    =    0,      j=1,2,3.

We must prove that p=0. 
By Lemma <ref>, we deduce that conditions (<ref>) and (<ref>) are equivalent to
 
    ℐ_j^CR(p)    =    0,      j=1,2,3, 
    ℒ^enr_j(p)    =    0,      j=1,2,3,
 
where ℒ^enr_j, j=1,2,3, is defined in (<ref>). 
The desired result can be directly inferred from the fact that the triple AF3, defined in (<ref>), is a finite element, as established in <cit.>.


As a direct consequence of Theorem <ref>, we can establish the existence of a basis

    ℬ_𝒞_α={ψ_i, ζ_i  :  i=1,2,3}

of ℙ_2(T) satisfying 
the following conditions <cit.>

    ℐ_j^CR(ψ_i) = δ_ij,   ℱ^enr_j,α(ψ_i) = 0,      i,j=1,2,3, 
          ℐ_j^CR(ζ_i) = 0,   ℱ^enr_j,α(ζ_i) = δ_ij,      i,j=1,2,3.

These functions, known as the basis functions of ℙ_2(T) associated to the enriched finite element 𝒞_α, are explicitly expressed in the following theorem.
 
Let α>-1 be a fixed real number different from -6/7. The basis functions ψ_i,ζ_i,  i=1,2,3, of ℙ_2(T) associated to the finite element 𝒞_α have the following expressions

    ψ_i=c_αφ_i +d_α∑_k=1 
     k ≠ i^3φ_k + ϕ_i,      ζ_i=1/γ_α L_α((K_α+h_α)φ_i-h_α∑_k=1 
     k ≠ i^3φ_k ),      i=1,2,3,

where

    c_α=3 (11 α^2  + 20 α +12 )/(α+6) ( 7 α+6),      d_α=3(- 3 α^2  + 8 α +12 ) /(α+ 6 ) (7 α+6 ),      
        L_α=-(α+ 6 ) (7 α+6 )/8 (2 α+3)^2,

and γ_α,h_α,K_α,φ_i,ϕ_i are defined in (<ref>) (<ref>) and (<ref>), respectively. 
      

   Without loss on generality, we prove the theorem for i=1. 

Since ψ_1∈ℙ_2(T), according to (<ref>), it can be expressed as

    ψ_1=∑_k=1^3 ℒ^enr_k(ψ_1) φ_k+∑_k=1^3 ℐ^CR_k(ψ_1) ϕ_k.

Using (<ref>) and (<ref>), we obtain

    ψ_1=∑_k=1^3 ψ_1( v_k) φ_k+ϕ_1.

Now, let us compute the vector

    [ψ_1( v_1),ψ_1( v_2),ψ_1( v_3)]^T.

For this purpose, we apply the operators ℱ^enr_j,α, j=1,2,3, to both sides of (<ref>), leveraging (<ref>), resulting in the following 
    
    0     =   ψ_1( v_1) ℱ_1,α^enr(φ_1) + ψ_1( v_2)ℱ_1,α^enr(φ_2)+  ψ_1( v_3)ℱ_1,α^enr(φ_3)+ℱ_1,α^enr(ϕ_1) 
    
          0   =   ψ_1( v_1)  ℱ_2,α^enr(φ_1)+ψ_1( v_2)ℱ_2,α^enr(φ_2)+ ψ_1( v_3) ℱ_2,α^enr(φ_3)+ℱ_2,α^enr(ϕ_1)  
    
           0    =   ψ_1( v_1) ℱ_3,α^enr(φ_1) + ψ_1( v_2) ℱ_3,α^enr(φ_2) +ψ_1( v_3)ℱ_3,α^enr(φ_3)+ℱ_3,α^enr(ϕ_1).

Substituting (<ref>) and (<ref>) into these equations, we obtain the linear system

    γ_α[ K_α h_α h_α; h_α K_α h_α; h_α h_α K_α ][ ψ_1( v_1); ψ_1( v_2); ψ_1( v_3) ]
    = -[ ℱ_1,α^enr(ϕ_1); ℱ_2,α^enr(ϕ_1); ℱ_3,α^enr(ϕ_1) ].

Thus, performing a straightforward calculation, we find
 
    -1/γ_αL_α[ K_α+h_α    -h_α    -h_α;    -h_α K_α+h_α    -h_α;    -h_α    -h_α K_α+h_α ][ ℱ_1,α^enr(ϕ_1); ℱ_2,α^enr(ϕ_1); ℱ_3,α^enr(ϕ_1) ]
    =  [ ψ_1( v_1); ψ_1( v_2); ψ_1( v_3) ].

By (<ref>), (<ref>) and (<ref>), we get 

    ℱ^enr_1,α(ϕ_1)=3(α+2)/2(2α+3)γ_α,     ℱ^enr_2,α(ϕ_1)= ℱ^enr_3,α(ϕ_1)=3(α+1)/2α+3γ_α.

Substituting (<ref>) into the linear system (<ref>) and solving it, we obtain 

    ψ_1( v_1)=3 (11 α^2  + 20 α +12 )/(α+6) ( 7 α+6),     ψ_1( v_2)=ψ_1( v_3)=3(- 3 α^2  + 8 α +12 ) /(α+ 6 ) (7 α+6 ).

Combining (<ref>) with (<ref>) and following the notations defined in (<ref>), we find

    ψ_1=c_αφ_1+d_α(φ_2+φ_3) + ϕ_1.



It remains to prove the expression for ζ_1. Since ζ_1∈ℙ_2(T), by (<ref>), it can be expressed as 

    ζ_1=∑_k=1^3 ℒ^enr_k(ζ_1) φ_k+∑_k=1^3 ℐ^CR_k(ζ_1) ϕ_k.

Using (<ref>) and (<ref>),  we obtain

    ζ_1=∑_k=1^3 ζ_1( v_j) φ_k.

Now, let us compute the vector

    [ζ_1( v_1),ζ_1( v_2),ζ_1( v_3)]^T.

For this purpose, leveraging (<ref>) of Lemma <ref> and (<ref>), we get
 
    γ_α[ K_α h_α h_α; h_α K_α h_α; h_α h_α K_α ][ ζ_1( v_1); ζ_1( v_2); ζ_1( v_3) ]
    = [ 1; 0; 0 ].

Thus, we find
 
    1/γ_αL_α[ K_α+h_α    -h_α    -h_α;    -h_α K_α+h_α    -h_α;    -h_α    -h_α K_α+h_α ][ 1; 0; 0 ]
    =  [ ζ_1( v_1); ζ_1( v_2); ζ_1( v_3) ],

and then

    ζ_1( v_1)=K_α+h_α/γ_α L_α,     ζ_1( v_2)=-h_α/γ_α L_α,     ζ_1( v_3)=-h_α/γ_α L_α.

Substituting these values into (<ref>), we have 

    ζ_1=1/γ_αL_α[(K_α+h_α)φ_1-h_αφ_2-h_αφ_3].

Analogously, the theorem can be established for i=2 and i=3; consequently, the thesis follows.






 Let α>-1 be a fixed real number different from -6/7. The approximation operator relative to the enriched finite element 𝒞_α

    [                     Π_2,𝒞_α^enr: C(T)                                     →                                ℙ_2(T);                                     f                                     ↦ ∑_j=1^3ℐ^CR_j(f)ψ_j+ ℱ^enr_j,α(f)ζ_j, ]

reproduces all polynomials of ℙ_2(T) and satisfies 

    ℐ^CR_j(Π_2,𝒞_α^enr[f])   =   ℐ^CR_j(f),      j=1,2,3, 
    ℱ^enr_j,α(Π_2,𝒞_α^enr[f])   =   ℱ^enr_j,α(f),      j=1,2,3.
 


The proof is a consequence of (<ref>) and (<ref>).










§ A SECOND ONE-PARAMETER FAMILY OF ENRICHMENT OF THE CROUZEIX–RAVIART FINITE ELEMENT


In this section, we introduce a new parametric family of quadratic enrichments of the Crouzeix–Raviart finite element. To achieve this, we introduce a real parameter β > -1. Using this parameter, we define the following enriched linear functionals 
 
    𝒢_j,β^enr(f) = ∫_0^1w_β(t)f(t v_j+ (1-t)
      m^⋆)  dt,      j=1,2,3,
 
 where 
 
    m^⋆= v_1+ v_2+ v_3/3

is the center of gravity of T and w_β(t)=t^β(1-t)^1-β.
The one-parameter family of triples is then defined as
 
    ℰ_β= (T,  ℙ_2(T),Σ_T,β^enr),

where   
 
    Σ_T,β^enr=  {ℐ_j^CR,𝒢^enr_j,β  :   j=1,2,3},

is the set of enriched degrees of freedom.  In the following, we establish that for any β>-1, the triple ℰ_β, defined in (<ref>), is a finite element. To prove this, we first introduce some preliminary results. By using the same notations of the previous section, we denote by

    ν_β = B(β+2,β+1)/3(2β+3),     σ_β=3β+4.


        We observe that, as a consequence of (<ref>) and (<ref>), we obtain
 
    λ_i( m^⋆)=1/3,      i=1,2,3.
 

 

  Let p ∈ℙ_2(T) such that ℐ_j^CR(p)=0, j=1,2,3. Then, the values of p at the vertices of T satisfy the following linear system 

    ν_β[    2 -σ_β -σ_β; -σ_β    2 -σ_β; -σ_β -σ_β    2 ][ p( v_1); p( v_2); p( v_3) ]
    = [ 𝒢_1,β^enr(p); 𝒢_2,β^enr(p); 𝒢_3,β^enr(p) ].

  


 Let p ∈ℙ_2(T) such that
 
    ℐ_j^CR(p)=0,      j=1,2,3.

Referring to (<ref>) and (<ref>), we can express p as follows 
 
    p=∑_k=1^3ℒ^enr_k(p)φ_k+∑_k=1^3 ℐ^CR_k(p) ϕ_k= ∑_k=1^3 p( v_k)φ_k.

Applying the functional 𝒢_j,β^enr, j=1,2,3, to both sides of (<ref>), we obtain 

    𝒢_j,β^enr(p) = ∑_k=1^3 p(v_k)𝒢_j,β^enr(φ_k),      j=1,2,3.

Using (<ref>) in combination with (<ref>) and (<ref>), and leveraging the properties of the Euler beta function (<ref>) and (<ref>), straightforward calculations lead to the following results

    𝒢^enr_j,β(φ_j)   =   2ν_β,      j=1,2,3,
    
    𝒢^enr_j,β(φ_k)   =   -σ_βν_β,      j,k=1,2,3,    j≠ k.

By substituting (<ref>) and (<ref>) into (<ref>), the thesis follows.






  Let p∈ℙ_2(T) such that ℐ_j^CR(p)=0, j=1,2,3. Then p vanishes at the vertices of the triangle T if and only if

    𝒢_j,β^enr(p)= 0,    j=1,2,3.

 
  
  Let p∈ℙ_2(T) such that 
   
    ℐ_j^CR(p) = 0,      j=1,2,3.

 By leveraging (<ref>) of Lemma <ref>, it follows that, if p vanishes at the vertices of the triangle T, then condition (<ref>) is satisfied. 
 It remains to prove the reverse implication. Then we assume that p satisfies condition (<ref>). In this hypothesis, the linear system (<ref>) becomes
 
    ν_β[    2 -σ_β -σ_β; -σ_β    2 -σ_β; -σ_β -σ_β    2 ][ p( v_1); p( v_2); p( v_3) ]
    = [ 0; 0; 0 ],

where ν_β,σ_β are defined in (<ref>).
The required result follows since the determinant of the matrix associated to the above linear system is 

    Δ_β=-54ν_β^3 (β+1)(β+2)^2

 which is different from zero for any β>-1.
 


  
 For any real number β > -1 the triple ℰ_β is a finite element. 


  Let p∈ℙ_2(T) such that

    ℐ_j^CR(p)    =    0,      j=1,2,3, 
    𝒢_j,β^enr(p)    =    0,      j=1,2,3.

We must prove that p=0. 
By Lemma <ref>, we deduce that conditions (<ref>) and (<ref>) are equivalent to
 
    ℐ_j^CR(p)    =    0,      j=1,2,3, 
    ℒ^enr_j(p)    =    0,      j=1,2,3,
 
where ℒ^enr_j, j=1,2,3, is defined in (<ref>). 
The desired result can be directly inferred from the fact that the triple AF3, defined in (<ref>), is a finite element, as established in <cit.>.
 


As a direct consequence of Theorem <ref>, we can establish the existence of a basis

    ℬ_ℰ_β={τ_i, ρ_i  :  i=1,2,3}

of ℙ_2(T) satisfying 
the following conditions <cit.>

    ℐ_j^CR(τ_i) = δ_ij,   𝒢^enr_j,β(τ_i) = 0,      i,j=1,2,3, 
          ℐ_j^CR(ρ_i) = 0,   𝒢^enr_j,β(ρ_i) = δ_ij,      i,j=1,2,3.

These functions, known as the basis functions of ℙ_2(T) associated to the enriched finite element ℰ_β, are explicitly expressed in the following theorem.




Let β>-1 be a fixed real number. The basis functions τ_i,ρ_i,  i=1,2,3, of ℙ_2(T) associated to the finite element ℰ_β have the following expressions

    τ_i=r_βφ_i+m_β∑_k=1 
     k ≠ i^3φ_k+ϕ_i,      ρ_i= 1/ν_βL_β((σ_β-2)φ_i-σ_β∑_k=1 
     k ≠ i^3φ_k ),      i=1,2,3,

where 

    r_β=6(7 β^2 + 18 β + 12)/ L_β,        m_β=6(β^2 + 6 β + 6)/ L_β,     
      L_β=18(β+1)(β+2)

and ν_β, σ_β,φ_i,ϕ_i are defined in (<ref>) (<ref>) and (<ref>), respectively. 
      

   Without loss on generality, we prove the theorem for i=1. 

Since τ_1∈ℙ_2(T), according to (<ref>), it can be expressed as

    τ_1=∑_k=1^3 ℒ^enr_k(τ_1) φ_k+∑_k=1^3 ℐ^CR_k(τ_1) ϕ_k.

Using (<ref>) and (<ref>), we obtain

    τ_1=∑_k=1^3 τ_1( v_k) φ_k+ϕ_1.

Now, let us compute the vector

    [τ_1( v_1),τ_1( v_2),τ_1( v_3)]^T.

For this purpose, we apply the operators 𝒢^enr_j,β, j=1,2,3, to both sides of (<ref>), leveraging (<ref>), resulting in the following 
    
    0     =   τ_1( v_1) 𝒢_1,β^enr(φ_1) + τ_1( v_2)𝒢_1,β^enr(φ_2)+  τ_1( v_3)𝒢_1,β^enr(φ_3)+𝒢_1,β^enr(ϕ_1) 
    
         0     =   τ_1( v_1) 𝒢_2,β^enr(φ_1) + τ_1( v_2)𝒢_2,β^enr(φ_2)+  τ_1( v_3)𝒢_2,β^enr(φ_3)+𝒢_2,β^enr(ϕ_1)
    
           0     =   τ_1( v_1) 𝒢_3,β^enr(φ_1) + τ_1( v_2)𝒢_3,β^enr(φ_2)+  τ_1( v_3)𝒢_3,β^enr(φ_3)+𝒢_3,β^enr(ϕ_1).

Substituting (<ref>) and (<ref>) into these equations, we obtain the linear system


    ν_β[    2 -σ_β -σ_β; -σ_β    2 -σ_β; -σ_β -σ_β    2 ][ τ_1( v_1); τ_1( v_2); τ_1( v_3) ]
    = -[ 𝒢_1,β^enr(ϕ_1); 𝒢_2,β^enr(ϕ_1); 𝒢_3,β^enr(ϕ_1) ].

Thus, performing a straightforward calculation, we find
 
    - 1/ν_βL_β[ σ_β-2  -σ_β  -σ_β;  -σ_β σ_β-2  -σ_β;  -σ_β  -σ_β σ_β-2 ][ 𝒢_1,β^enr(ϕ_1); 𝒢_2,β^enr(ϕ_1); 𝒢_3,β^enr(ϕ_1) ]
    =  [ τ_1( v_1); τ_1( v_2); τ_1( v_3) ].

By (<ref>), (<ref>) and (<ref>), we get 

    𝒢^enr_1,β(ϕ_1)=2(β+2) ν_β,     𝒢^enr_2,β(ϕ_1)= 𝒢^enr_3,β(ϕ_1)=2(4β+5)ν_β.

Substituting (<ref>) into the linear system (<ref>) and solving it, we obtain 

    τ_1( v_1)=7 β^2 + 18 β + 12/3 β^2 + 9 β + 6,     τ_1( v_2)=τ_1( v_3)=β^2 + 6 β + 6 /3 β^2 + 9 β + 6.

Combining (<ref>) with (<ref>) and following the notations defined in (<ref>), we find

    τ_1=r_βφ_1+m_β(φ_2+φ_3) + ϕ_1.


It remains to prove the expression for ρ_1. Since ρ_1∈ℙ_2(T), by (<ref>), it can be expressed as 

    ρ_1=∑_k=1^3 ℒ^enr_k(ρ_1) φ_k+∑_k=1^3 ℐ^CR_k(ρ_1) ϕ_k.

Using (<ref>) and (<ref>),  we obtain

    ρ_1=∑_k=1^3 ρ_1( v_k) φ_k.

Now, let us compute the vector

    [ρ_1( v_1),ρ_1( v_2),ρ_1( v_3)]^T.

For this purpose, leveraging (<ref>) of Lemma <ref> and (<ref>), we get
 
    ν_β[    2 -σ_β -σ_β; -σ_β    2 -σ_β; -σ_β -σ_β    2 ][ ρ_1( v_1); ρ_1( v_2); ρ_1( v_3) ]
    = [ 1; 0; 0 ].

Thus, we find
 
    1/ν_βL_β[ σ_β-2  -σ_β  -σ_β;  -σ_β σ_β-2  -σ_β;  -σ_β  -σ_β σ_β-2 ][ 1; 0; 0 ]
    =  [ ρ_1( v_1); ρ_1( v_2); ρ_1( v_3) ],

and then

    ρ_1( v_1)=σ_β-2/ν_β L_β,   ρ_1( v_2)=-σ_β/ν_βL_β,   ρ_1( v_3)=-σ_β/ν_βL_β.

Substituting these values into (<ref>), we have 

    ρ_1=1/ν_β L_β[(σ_β-2)φ_1-σ_β(φ_2+φ_3)].

Analogously, the theorem can be established for i=2 and i=3; consequently, the thesis follows.



 Let β>-1 be a fixed real number. The approximation operator relative to the enriched finite element ℰ_β

    [                     Π_2,ℰ_β^enr: C(T)                                     →                                ℙ_2(T);                                     f                                     ↦ ∑_j=1^3ℐ^CR_j(f)τ_j+ 𝒢^enr_j,β(f)ρ_j, ]

reproduces all polynomials of ℙ_2(T) and satisfies 

    ℐ^CR_j(Π_2,ℰ_β^enr[f])   =   ℐ^CR_j(f),      j=1,2,3, 
    𝒢^enr_j,β(Π_2,ℰ_β^enr[f])   =   𝒢^enr_j,β(f),      j=1,2,3.
 


The proof is a consequence of (<ref>) and (<ref>).






§ NUMERICAL EXPERIMENTS

In this Section,  we test the effectiveness of the proposed enrichment strategies through multiple examples. We consider the following test functions used in <cit.>

    f_1(x,y)=sin(2π x)cos(2π y)/2,      f_2(x,y)=1/x^2+y^2+8,
            f_3(x,y)= e^-81/16((x-0.5)^2+(y-0.5)^2)/3, 
           f_4(x,y)=√(64-81((x-0.5)^2+(y-0.5)^2))/9-0.5,
           f_5(x,y)=e^x+y,      f_6(x,y)=1/x^2+y^2+25,

and six different Delaunay triangulations (see Figure <ref> and Figure <ref>). These latter are obtained through the Shewchuk's triangle program <cit.>. 

Numerical results are reported in Tables <ref>-<ref>. In these tables, we compare the error in L^1-norm produced by the standard Crouzeix–Raviart finite element (E^CR_1) with that produced by the enriched approximation operators Π_2,𝒞_α^enr, defined in (<ref>) with α=1 (E^enr_𝒞_α,1), and Π_2,ℰ_β^enr, defined in (<ref>) with β=1 (E^enr_ℰ_β,1).

As evident, the approximation generated by the new enriched finite elements outperforms the approximation provided by the traditional Crouzeix–Raviart finite element. This enhancement becomes particularly noteworthy as the number of triangles in a triangulation increases.



 

 






















§ ACKNOWLEDGMENTS

This research has been achieved as part of RITA “Research
 ITalian network on Approximation” and as part of the UMI group “Teoria dell'Approssimazione
 e Applicazioni”. The author is a member of the INdAM Research group GNCS.

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