Mixed Finite Element Methods

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Ricardo G. Dur´an

Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pabell´on I, 1428 Buenos Aires, Argentina

rduran@dm.uba.ar

1 Introduction

Finite element methods in which two spaces are used to approximate two different variables receive the general denomination of mixed methods. In some cases, the second variable is introduced in the formulation of the problem because of its physical interest and it is usually related with some derivatives of the original variable. This is the case, for example, in the elasticity equations, where the stress can be introduced to be approximated at the same time as the displacement. In other cases there are two natural independent variables and so, the mixed formulation is the natural one. This is the case of the Stokes equations, where the two variables are the velocity and the pressure.

The mathematical analysis and applications of mixed finite element methods have been widely developed since the seventies. A general analysis for this kind of methods was first developed by Brezzi [13]. We also have to mention the papers by Babuˇska [9] and by Crouzeix and Raviart [22] which, although for particular problems, introduced some of the fundamental ideas for the analysis of mixed methods.

We also refer the reader to [32, 31], where general results were obtained, and to the books [17, 45, 37].

The rest of this work is organized as follows: in Sect. 2 we review some basic tools for the analysis of finite element methods. Section 3 deals with the mixed formulation of second order elliptic problems and their finite element approximation.

We introduce the Raviart–Thomas spaces [44, 49, 41] and their generalization to higher dimensions, prove some of their basic properties, and construct the Raviart–

Thomas interpolation operator which is a basic tool for the analysis of mixed methods. Then, we prove optimal order error estimates and a superconvergence result for the scalar variable. We follow the ideas developed in several papers (see for example [24, 16]). Although for simplicity we consider the Raviart–Thomas spaces, the error analysis depends only on some basic properties of the spaces and the interpolation operator, and therefore, analogous results hold for approximations obtained with other finite element spaces. We end the section recalling other known fami- lies of spaces and giving some references. In Sect. 4 we introduce an a posteriori

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error estimator and prove its equivalence with an appropriate norm of the error up to higher order terms. For simplicity, we present the a posteriori error analysis only in the 2-d case. Finally, in Sect. 5, we introduce the general abstract setting for mixed formulations and prove general existence and approximation results.

2 Preliminary Results

In this section we recall some basic results for the analysis of finite element approximations.

We will use the standard notation for Sobolev spaces and their norms, namely, given a domain Ω⊂ IRⁿand any positive integer k

H^k(Ω) ={φ ∈ L²(Ω) : D^αφ∈ L²(Ω) ∀ |α| ≤ k}, where

α = (α₁,· · · , αn), |α| = α1+· · · + αn and D^αφ = ∂^|α|φ

∂x^α₁¹· · · ∂x^αnⁿ

and the derivatives are taken in the distributional or weak sense.

H^k(Ω) is a Hilbert space with the norm given by

φ²_H^k_(Ω)=

|α|≤k

D^αφ²_L²_(Ω).

Given φ∈ H^k(Ω) and j∈ IN such 1 ≤ j ≤ k we define ∇^jφ by

|∇^jφ|²=

|α|=j

|D^αφ|².

Analogous notations will be used for vector fields, i.e., if v = (v1,· · · , vn) then D^αv = (D^αv₁,· · · , D^αv_n) and

v²_H^k_(Ω)=

n i=1

vi²_H^k_(Ω) and |∇^jv|²=

n i=1

|∇^jv_i|².

We will also work with the following subspaces of H¹(Ω):

H₀¹(Ω) ={φ ∈ H¹(Ω) : φ|∂Ω = 0}, H¹(Ω) ={φ ∈ H¹(Ω) :

Ω

φ dx = 0}.

Also, we will use the standard notationPkfor the space of polynomials of degree less than or equal to k and, if x ∈ IRⁿ and α is a multi-index, we will set x^α = x^α₁¹· · · x^αnⁿ.

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The letter C will denote a generic constant not necessarily the same at each occurrence.

Given a function in a Sobolev space of a domain Ω it is important to know whether it can be restricted to ∂Ω, and conversely, when can a function defined on

∂Ω be extended to Ω in such a way that it belongs to the original Sobolev space. We will use the following trace theorem. We refer the reader for example to [38, 33] for the proof of this theorem and for the definition of the fractional-order Sobolev space H¹²(∂Ω).

Theorem 2.1. Given φ∈ H¹(Ω), where Ω⊂ IRⁿis a Lipschitz domain, there exists a constant C depending only on Ω such that

φ_H¹₂_(∂Ω)≤ CφH¹(Ω). In particular,

φL²(∂Ω)≤ CφH¹(Ω). (1) Moreover, if g∈ H¹²(∂Ω), there exists φ∈ H¹(Ω) such that φ|∂Ω = g and

φH¹(Ω)≤ Cg

H¹²(∂Ω).

One of the most important results in the analysis of variational methods for elliptic problems is the Friedrichs–Poincar´e inequality for functions with vanishing mean average, that we state below (see for example [36] for the case of Lipschitz domains and [43] for another proof in the case of convex domains). Assume that Ω is a Lip- schitz domain. Then, there exists a constant C depending only on the domain Ω such that for any f ∈ H¹(Ω),

fL²(Ω)≤ C∇fL²(Ω). (2) The Friedrichs–Poincar´e inequality can be seen as a particular case of the next result on polynomial approximation which is basic in the analysis of finite element methods.

Several different arguments have been given for the proof of the next lemma. See for example [12, 25, 26, 51]. Here we give a nice argument which, to our knowledge, is due to M. Dobrowolski for the lowest order case on convex domains (and as far as we know has not been published). The proof given here for the case of domains which are star-shaped with respect to a subset of positive measure and any degree of approximation is an immediate extension of Dobrowolski’s argument. For simplicity we present the proof for the L²-case (which is the case that we will use), but the reader can check that an analogous argument applies for L^p based Sobolev spaces (1≤ p < ∞).

Assume that Ω is star-shaped with respect to a set B⊆ Ω of positive measure.

Given an integer k ≥ 0 and f ∈ H^k+1(Ω) we introduce the averaged Taylor poly- nomial approximation of f , Qk,Bf ∈ Pkdefined by

Qk,Bf (x) = 1

|B|

B

Tkf (y, x) dy

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where T_kf (y, x) is the Taylor expansion of f centered at y, namely, Tkf (y, x) =

|α|≤k

D^αf (y)(x− y)^α α! .

Lemma 2.1. Let Ω ⊂ IRⁿ be a domain with diameter d which is star-shaped with respect to a set of positive measure B ⊂ Ω. Given an integer k ≥ 0 and f ∈ H^k+1(Ω), there exists a constant C = C(k, n) such that, for 0≤ |β| ≤ k + 1,

D^β(f− Qk,Bf )L²(Ω)≤ C|Ω|^1/2

|B|^1/2d^k+1−|β|∇^k+1fL²(Ω). (3) In particular, if Ω is convex,

D^β(f− Qk,Ωf )L²(Ω)≤ C d^k+1^−|β|∇^k+1fL²(Ω). (4)

Proof. By density we can assume that f ∈ C^∞(Ω). Then we can write

f (x)− Tkf (y, x) = (k + 1)

|α|=k+1

(x− y)^α α!

₁

0

D^αf (ty + (1− t)x) t^kdt.

Integrating this inequality over B (in the variable y) and dividing by|B| we have

f (x)− Qk,Bf (x) = k + 1

|B|

|α|=k+1

B

₁

0

(x− y)^α

α! D^αf (ty + (1− t)x) t^kdt dy and so,

Ω

|f(x) − Qk,Bf (x)|²dx

≤ C d^2(k+1)

|B|²

|α|=k+1

Ω

B

₁

0

|D^αf (ty + (1− t)x)|t^kdt dy

2

dx

≤ C d^2(k+1)

|B|²

|α|=k+1

Ω

B

₁

0

|D^αf (ty +(1− t)x)|²dt dy

B

₁

0

t^2kdt dy

dx.

Therefore,

Ω

|f(x)−Qk,Bf (x)|²dx

≤ Cd^2(k+1)

|B|

|α|=k+1

Ω

B

1 0

|D^αf (ty + (1− t)x)|²dt dy dx.

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Now, for each α,

Ω

B

₁

0

|D^αf (ty + (1− t)x)|²dt dy dx

=

Ω

B

¹

2

0

|D^αf (ty + (1− t)x)|²dt dy dx

+

Ω

B

₁

1 2

|D^αf (ty + (1− t)x)|²dt dy dx =: I + II.

Let us call g_αthe extension by zero of D^αf to IRⁿ. Then, by Fubini’s theorem and two changes of variables we have

I≤

B

¹

2

0

IRⁿ

|gα(ty +(1−t)x)|²dx dt dy =

B

¹

2

0

IRⁿ

|gα((1−t)x)|²dx dt dy

=

B

¹₂

0

IRⁿ

|gα(z)|²(1− t)⁻ⁿdz dt dy≤ 2ⁿ⁻¹|B|

Ω

|D^αf (z)|²dz.

Analogously,

II ≤

Ω

₁

1 2

IRⁿ

|gα(ty + (1− t)x)|²dy dt dx =

Ω

₁

1 2

IRⁿ

|gα(ty)|²dy dt dx

=

Ω

₁

1 2

IRⁿ

|gα(z)|²t⁻ⁿdz dt dx≤ 2ⁿ⁻¹|Ω|

Ω

|D^αf (z)|²dz.

Therefore, replacing these bounds in (5) we obtain (3) for β = 0.

On the other hand, an elementary computation shows that D^βQ_k,Bf (x) = Q_k−|β|,B(D^βf )(x) ∀|β| ≤ k

and therefore, the estimate (3) for|β| > 0 follows from the case β = 0 applied to D^βf .

Important consequences of this result are the following error estimates for the L²-projection ontoPm.

Corollary 2.1. Let Ω ⊂ IRⁿ be a domain with diameter d star-shaped with respect to a set of positive measure B⊂ Ω. Given an integer m ≥ 0, let P : L²(Ω)→ Pm

be the L²-orthogonal projection. There exists a constant C = C(j, n) such that, for 0≤ j ≤ m + 1, if f ∈ H^j(Ω), then

f − P fL²(Ω)≤ C|Ω|^1/2

|B|^1/2d^j|∇^jf|L²(Ω).

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Remark 2.1. Analogous results to Lemma 3 and its corollary hold for bounded Lip- schitz domains because this kind of domains can be written as a finite union of star- shaped domains (see [25] for details).

The following result is fundamental in the analysis of mixed finite element approximations.

Lemma 2.2. Let Ω ⊂ IRⁿ be a bounded domain. Given f ∈ L²(Ω) there exists v∈ H¹(Ω)ⁿsuch that

div v = f in Ω (6)

and

vH¹(Ω)≤ CfL²(Ω) (7)

with a constant C depending only on Ω.

Proof. Let B ∈ IRⁿ be a ball containing Ω and φ be the solution of the boundary

problem

∆φ = f in B

φ = 0 on ∂B (8)

It is known that φ satisfies the following a priori estimate (see for example [36])

φH²(Ω)≤ CfL²(Ω)

and therefore v =∇φ satisfies (6) and (7).

Remark 2.2. To treat Neumann boundary conditions we would need the existence of a solution of divv = f satisfying (7) and the boundary condition v· n = 0 on ∂Ω.

Such a v can be obtained by solving a Neumann problem in Ω for smooth domains or convex polygonal or polyhedral domains. For more general domains, including arbitrary polygonal or polyhedral domains, the existence of v satisfying (6) and (7) can be proved in different ways. In fact v can be taken such that all its components vanish on ∂Ω (see for example [2, 7, 30]).

A usual technique to obtain error estimates for finite element approximations is to work in a reference element and then change variables to prove results for a general element. Let us introduce some notations and recall some basic estimates.

Fix a reference simplex T ⊂ IRⁿ. Given a simplex T ⊂ IRⁿ, there exists an invertible affine map F : T → T , F (ˆx) = Aˆx + b, with A ∈ IR^n×nand b∈ IRⁿ.

We call hT the diameter of T and ρT the diameter of the largest ball inscribed in T (see Fig. 1). We will use the regularity assumption on the elements, namely, many of our estimates will depend on a constant σ such that

hT

ρ_T ≤ σ. (9)

It is known that (see [19]), for the matrix norm associated with the euclidean vector norm, the following estimates hold:

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h_T

F

h_T

ˆ

ρ_T^ˆ ρ_T

Fig. 1.

A ≤ h_T

ρ^T and A⁻¹ ≤ hT

ρT

. (10)

With any φ∈ L²(T ) we associate ˆφ∈ L²( T ) in the usual way, namely,

φ(x) = ˆφ(ˆx) (11)

where x = F (ˆx).

We end this section by recalling the so-called inverse estimates which are a fundamental tool in finite element analysis. We give only a particular case which will be needed for our proofs (see for example [19] for more general inverse estimates).

Lemma 2.3. Given a simplex T there exists a constant C = C(σ, k, n, T ) such that, for any p∈ Pk(T ),

∇pL²(T )≤ C

hTpL²(T ).

Proof. SincePk( T ) is a finite-dimensional space, all the norms defined on it are equivalent. In particular, there exists a constant C depending on k and T such that

 ∇ˆp_L2(^{T )} ≤ Cˆp_L2(^{T )} (12) for any ˆp∈ Pk( T ).

An easy computation shows that

∇p = A^−T∇ˆp

where A^−T is the transpose matrix of A⁻¹. Therefore, using the bound forA⁻¹ given in (10) together with (12) and (9) we have

T

|∇p|²dx =

T

|A^−T∇ˆp| ²| det A| dˆx ≤ A⁻¹²

T

| ∇ˆp|²| det A| dˆx

≤ C h²T ρ²_T

T

|ˆp|²| det A| dˆx = C h²T ρ²_T

T

|p|²dx≤ Cσ² h²T h²_T

T

|p|²dx.

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3 Mixed Approximation of Second Order Elliptic Problems

In this section we introduce the mixed finite element approximation of second order elliptic problems and we develop the a priori error analysis. We consider the so called h-version of the finite element method, namely, fixing a degree of approximation we prove error estimates in terms of the mesh size. We present the error analysis for the case of L²based norms (following essentially [24]) and refer to [27, 34, 35] for error estimates in other norms.

As it is usually done, we prove error estimates for any degree of approximation under the hypothesis that the solution is regular enough in order to show the best possible order of a method. However, the reader has to be aware that, in practice, for polygonal or polyhedral domains (which is the case considered here!) the solution is in general not smooth due to singularities at the angles and therefore the order of convergence is limited by the regularity of the solution of each particular problem considered. On the other hand, for domains with smooth boundary where the solutions might be very regular, a further error analysis considering the approximation of the boundary is needed.

Consider the elliptic problem

− div(a∇p) = f in Ω

p = 0 on ∂Ω (13)

where Ω⊂ IRⁿis a polyhedral domain and a = a(x) is a function bounded by above and below by positive constants.

In many applications the variable of interest is u =−a∇p

and then, it could be desirable to use a mixed finite element method which approxi- mates u and p simultaneously. With this purpose, problem (13) is decomposed into a first order system as follows:

⎧⎨

⎩

u + a∇p = 0 in Ω div u = f in Ω

p = 0 on ∂Ω.

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To write an appropriate weak formulation of this problem we introduce the space H(div, Ω) ={v ∈ L²(Ω)ⁿ: div v∈ L²(Ω)}

which is a Hilbert space with norm given by

v²_H(div,Ω)=v²_L²_(Ω)+ div v²_L²_(Ω). Defining µ(x) = 1/a(x), the first equation in (14) can be rewritten as

µ u +∇p = 0 in Ω.

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Multiplying by test functions and integrating by parts we obtain the standard weak mixed formulation of problem (14), namely,

⎧⎪

⎪⎨

⎪⎪

⎩

Ω

µu· v dx −

Ω

p div v dx = 0 ∀v ∈ H(div, Ω)

Ω

q div u dx =

Ω

f q dx ∀q ∈ L²(Ω).

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Observe that the Dirichlet boundary condition is implicit in the weak formulation (i.e., it is the type of condition usually called natural). Instead, Neumann boundary conditions would have to be imposed on the space (essential conditions). This is exactly opposite to what happens in the case of standard formulations.

The weak formulation (15) involves the divergence of the solution and of the test functions but not arbitrary first derivatives. This fact allows us to work on the space H(div, Ω) instead of the smaller H¹(Ω)ⁿ and this will be important for the finite element approximation because piecewise polynomials vector functions do not need to have both components continuous to be in H(div, Ω), but only their normal component.

In order to define finite element approximations to the solution (u, p) of (15) we need to introduce finite-dimensional subspaces of H(div, Ω) and L²(Ω) made of piecewise polynomial functions.

For simplicity we will consider the case of triangular elements (or its generaliza- tions to higher dimensions) and the associated Raviart–Thomas spaces which are the best-known spaces for this problem. This family of spaces was introduced in [44]

in the 2-d case, while its extension to three dimensions was first considered in [41].

Since no essential technical difficulties arise in the general case, we prefer to present the spaces and the analysis of their properties in the general n-dimensional case (al- though, of course, we are mainly interested in the cases n = 2 and n = 3). Below we will comment and give references on different variants of spaces.

First we introduce the local spaces, analyze their properties and construct the Raviart–Thomas interpolation.

Given a simplex T ∈ IRⁿ, the local Raviart–Thomas space [44, 41] of order k≥ 0 is defined by

RTk(T ) =Pk(T )ⁿ+ xPk(T ) (16) In the following lemma we give some basic properties of the spacesRTk(T ).

We denote with Fi, i = 1,· · · , n + 1, the faces of a simplex T and with ni their corresponding exterior normals.

Lemma 3.1. (a)dimRTk(T ) = n_k+n

k

+_k+n₋₁

k

.

(b) If v∈ RTk(T ) then, v· ni∈ Pk(F_i) for i = 1,· · · , n + 1.

(c) If v∈ RTk(T ) is such that div v = 0 then, v∈ P_kⁿ. Proof. Any v∈ RTk(T ) can be written as

v = w + x

|α|=k

a_αx^α (17)

with w∈ P_kⁿ.

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Recall that dimPk = _k+n

k

and that the number of multi-indeces α such that

|α| = k is_k+n₋₁

k

. Then, (a) follows from (17).

Now, the face Fiis on a hyperplane of equation x·ni= s with s∈ IR. Therefore, if v = w + x p with w∈ P_kⁿand p∈ Pk, we have

v· ni= w· ni+ x· nip = w· ni+ s p∈ Pk

which proves (b).

Finally, if div v = 0 we take the divergence in the expression (17) and conclude easily that aα= 0 for all α and therefore (c) holds.

Our next goal is to construct an interpolation operator ΠT : H¹(T )ⁿ→ RTk

which will be fundamental for the error analysis. We fix k and to simplify notation we omit the index k in the operator.

For simplicity we define the interpolation for functions in H¹(T )ⁿalthough it is possible (and necessary in many cases!) to do the same construction for less regular functions. Indeed, the reader who is familiar with fractional order Sobolev spaces and trace theorems will realize that the degrees of freedom defining the interpolation are well defined for functions in H^s(T )ⁿ, with s > 1/2.

The local interpolation operator is defined in the following lemma.

Lemma 3.2. Given v∈ H¹(T )ⁿ, where T ∈ IRⁿis a simplex, there exists a unique ΠTv∈ RTk(T ) such that

Fi

ΠTv· nipkds =

Fi

v· nipkds ∀pk∈ Pk(Fi) , i = 1,· · · , n + 1 (18) and, if k≥ 1,

T

Π_Tv· pk−1dx =

T

v· pk−1dx ∀pk−1 ∈ Pkⁿ−1(T ). (19)

Proof. First, we want to see that the number of conditions defining ΠTv equals the dimension ofRTk(T ). This is easily verified for the case k = 0, so let us consider the case k≥ 1.

Since dimPk(Fi) =_k+n₋₁

k

, the number of conditions in (18) is

# of faces × dim Pk(F_i) = (n + 1)

k + n− 1 k

. On the other hand, the number of conditions in (19) is

dimPkⁿ−1(T ) = n

k + n− 1 k− 1

.

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Then, the total number of conditions defining Π_Tv is (n + 1)

k + n− 1 k

+ n

k + n− 1 k− 1

. Therefore, in view of (a) of lemma (3.1), we have to check that

n k + n

k

+

k + n− 1 k

= (n + 1)

k + n− 1 k

+ n

k + n− 1 k− 1

or equivalently,

k + n k

=

k + n− 1 k

+

k + n− 1 k− 1

which can be easily verified.

Therefore, in order to show the existence of ΠTv, it is enough to prove unique- ness. So, take v∈ RTk(T ) such that

Fi

v· nipkds = 0 ∀pk∈ Pk(Fi) , i = 1,· · · , n + 1 (20)

and

T

v· pk−1dx = 0 ∀pk−1∈ Pkⁿ−1(T ). (21) From (b) of Lemma 3.1 and (20) it follows that v· ni = 0 on Fi. Then, using now (21) we have

T

(div v)²dx =−

T

v· ∇(div v) dx = 0

because ∇(div v) ∈ P_k−1ⁿ (T ). Consequently div v = 0 and so, from (c) of Lemma 3.1 we know that v∈ P_kⁿ(T ).

Therefore, for each i = 1,· · · , n+1, the component v ·niis a polynomial of de- gree k on T which vanishes on F_i. Therefore, calling λ_ithe barycentric coordinates associated with T (i.e., λ_i(x) = 0 on F_i), we have

v· ni= λ_iq_k−1 with qk−1∈ Pk−1(T ). But, from (21) we know that

T

v· nip_k−1dx = 0 ∀pk−1∈ Pk−1(T ) and choosing pk−1 = qk−1we obtain

T

λ_iq_k−1² dx = 0.

Therefore, since λi does not change sign on T , it follows that q_k−1 = 0 and con- sequently v· ni = 0 in T for i = 1,· · · , n + 1. In particular, there are n linearly independent directions in which v has vanishing components and, therefore, v = 0 as we wanted to see.

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} >

?

}

} >

>

? ?

Fig. 2.Degrees of freedom forRT0andRT1in IR²

Figure 2 shows the degrees of freedom defining ΠT for k = 0 and k = 1 in the 2-d case. The arrows indicate normal components values and the filled circle, moments of the components of v (and so it corresponds to two degrees of freedom).

To obtain error estimates for the mixed finite element approximations we need to know the approximation properties of the Raviart–Thomas interpolation ΠT. The analysis given in [44, 49] makes use of general standard arguments for polynomial- preserving operators (see [19]). The main difference with the error analysis for La- grange interpolation is that here we have to use an appropriate transformation, known as the Piola transform, which preserves the degrees of freedom defining Π_Tv.

The Piola transform is defined as follows. Given two domains Ω, Ω ⊂ IRⁿ and a smooth bijective map F : Ω → Ω, let DF be the Jacobian matrix of F and J := det DF . Assume that J does not vanish at any point, then, we define for ˆ

v∈ L²( Ω)ⁿ

v(x) = 1

|J(ˆx)|DF (ˆ x)ˆv(ˆx)

where x = F (ˆx). Here and in what follows, the hat over differential operators indi- cates that the derivatives are taken with respect to ˆx.

We recall that scalar functions are transformed as indicated in (11) (we are using the same notation for the transformation of vector and scalar functions since no confusion is possible).

In the particular case that F is an affine map given by Aˆx + b we have J = det A and

v(x) = 1

|J|Aˆv(ˆx). (22)

In the next lemma we give some fundamental properties of the Piola transform.

For simplicity, we prove the results only for affine transformations, which is the use- ful case for our purposes. However, it is important to remark that analogous results hold for general transformations and this is important, for example, to work with general quadrilateral elements.

Lemma 3.3. If v∈ H(div, T ) and φ ∈ H¹(T ) then

T

div v φ dx =

T

div ˆv ˆφ dˆx , (23)

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T

v· ∇φ dx =

T

ˆ

v· ˆ∇ ˆφ dˆx (24)

and

∂T

v· n φ ds =

∂T ˆ

v· ˆn ˆφ dˆs. (25)

Proof. From the definition of the Piola transform (22) we have Dv(x) = 1

|J|AD(ˆv◦ F⁻¹)(x) = 1

|J|A Dˆv(ˆx)DF⁻¹(x) = 1

|J|A Dˆv(ˆx)A⁻¹. Then,

div v = tr Dv = 1

|J|tr(A DˆvA⁻¹) = 1

|J|tr Dˆv = 1

|J|divˆv and therefore (23) follows by a change of variable.

To prove (24) recall that

∇φ = A^−T∇ ˆφ.ˆ

Then,

T

v· ∇φ dx =

T

Aˆv· A^−T∇ ˆφ dˆx =ˆ

T

ˆ

v· ˆ∇ ˆφ dˆx.

Finally, (25) follows from (23) and (24) applying the divergence theorem.

Remark 3.1. The integral over ∂T in the previous lemma has to be understood as a duality product between v· n ∈ H⁻¹²(∂T ) and φ∈ H¹²(∂T ).

We can now prove the invariance of the Raviart–Thomas interpolation under the Piola transform.

Lemma 3.4. Given a simplex T ∈ IRⁿand v∈ H¹(T )ⁿwe have

ΠTˆv = ˆ ΠTv. (26)

Proof. We have to check that Π_Tv satisfies the conditions defining ΠTv, namely,ˆ

Fi

Π_Tv· ˆnipˆ_kdˆs =

Fi

ˆ

v· ˆnipˆ_kdˆs ∀ˆpk∈ Pk( F_i) , i = 1,· · · , n + 1 , (27)

where Fi= F⁻¹(Fi), and

T

ΠTv· ˆpk−1dˆx =

T

ˆ

v· ˆpk−1dˆx ∀ˆpk−1 ∈ P_k−1ⁿ ( T ). (28)

Given ˆpk ∈ Pk( Fi) we have

Fi

ˆ

v· ˆnipˆ_kdˆs =

Fi

v· nip_kds. (29)

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Indeed, this follows from (25) by a density argument. We can not apply (25) directly because the function obtained by extending p_k by zero to the other faces of T is not in H¹²(∂T ) and, therefore, it is not the restriction to the boundary of a function φ∈ H¹(T ). However, we can take a sequence of functions q_j ∈ C₀^∞(F_i) such that q_j→ pkin L²(F_i) and, since the extension by zero to ∂T of q_jis in H¹²(∂T ), there exists φj ∈ H¹(T ) such that the restriction of φj to Fi is equal to qj. Therefore, applying (25) we obtain,

Fi

ˆ

v· ˆniqˆjdˆs =

Fi

v· niqjds

and therefore, since v· ni∈ L²(Fi), we can pass to the limit to obtain (29). Analo- gously we have

Fi

ΠTv· ˆnipˆkdˆs =

Fi

ΠTv· nipkds.

and therefore (27) follows from condition (18) in the definition of Π_Tv.

To check (28) observe that, for ˆp_k−1∈ P_k−1ⁿ ( T ), we have

T

Π_Tv· ˆpk−1dˆx =

T

|J|A⁻¹Π_Tv· |J|A⁻¹p_k−1|J|⁻¹dx

=

T

Π_Tv· |J|A^−TA⁻¹p_k−1dx

=

T

v· |J|A^−TA⁻¹p_k−1dx =

T

ˆ

v· ˆpk−1dˆx where we have used condition (19) and that|J|A^−TA⁻¹pk−1∈ P_kⁿ₋₁(T ).

We can now prove the optimal order error estimates for the Raviart–Thomas interpolation.

Theorem 3.1. There exists a constant C depending on k, n and the regularity con- stant σ such that, for any v∈ H^m(T )ⁿand 1≤ m ≤ k + 1,

v − ΠTvL²(T )≤ Ch_T^m∇^mvL²(T ). (30)

Proof. First we prove an estimate on the reference element T . We will denote with C a generic constant which depends only on k, n and T . For each face Fiof T let {pⁱ_j}1≤j≤Nbe a basis ofPk( Fi) and let{pm}1≤m≤Mbe a basis ofP_kⁿ₋₁( T ). Then, associated with this basis we can introduce the Lagrange-type basis of RTk( T ), {φⁱ_j, ψm} defined by

Fi

φⁱ_j· nip^r_s= δirδjs,

T

φⁱ_j· pm= 0 ,

∀ i, r = 1,· · ·, n + 1, j, s = 1,· · ·, N, m = 1,· · ·, M

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and

T

ψ_m· p= δ_m, ψ_m· ni= 0

∀ m, = 1,· · ·, M, i = 1,· · ·, n + 1.

Then,

Π^Tv(ˆˆ x) =

n+1

i=1

N j=1

Fi

vˆ· nipⁱ_j

φⁱ_j(ˆx) +

M m=1

T

vˆ· pm

ψm(ˆx).

Now, from the trace theorem (1) on T we have

Fi

ˆ

v· nipⁱ_j ≤ Cˆv_H1(T ). Clearly, we also have

Tˆ

ˆ

v· pm ≤ Cˆv_L2(^{T )}.

In both estimates the constant C depends on bounds for the polynomials pⁱ_jand pm

and then, it depends only on k, n and T .

Therefore, using now thatφⁱ_j_L2(^{T )} andψm_L2(^{T )} are also bounded by a constant C we obtain

Π^Tvˆ_L2(T ) ≤ Cˆv_H1(T ). (31) Using now the relation (26) and making a change of variables we have

T

|ΠTv|²dx =

T

|J|⁻²|AΠTv|ˆ²|J| dˆx ≤ |J|⁻¹A²

T|ΠTv|ˆ ²dˆx.

Then, using the bound forA (10) and (31) we obtain

T

|ΠTv|²dx≤ |J|⁻¹h²_T ρ²T

T

|ˆv|²dˆx +

T

| Dˆv|²dˆx

(32)

but, since ˆv = |J|A⁻¹v and Dˆv = |J|A⁻¹DvA, using the bounds for A and

A⁻¹ (10),

|ˆv| ≤ |J|hT

ρT|v| and | Dˆv| ≤ |J|hT

ρT

h_T ρ^T|Dv|

and so, it follows from (32), changing variables again, that

ΠTv²_L²_{(T )}≤ C

h²_T

ρ²_Tv²_L²_{(T )}+h⁴_T

ρ²_TDv²_L²_{(T )} .

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Therefore, from the regularity hypothesis (9) we obtain

ΠTvL²(T )≤ C

vL²(T )+ hTDvL²(T )

(33) where the constant depends only on T , k, n and the regularity constant σ.

Now we use a standard argument. Since P_kⁿ(T ) ⊂ RTk(T ) we know that ΠTq = q for all q∈ P_kⁿ(T ) and then

v − ΠTvL²(T )=v − q − ΠT(v− q)L²(T )

≤ C{v − qL²(T )+ h_TD(v − q)L²(T )}

where the constant depends on that in (33). Therefore, we conclude the proof applying Lemma 2.1.

Let us now introduce the global Raviart–Thomas finite element spaces. Assume that we have a family of triangulations{Th} of Ω, i.e., Ω = ∪T∈ThT , such that the intersection of two elements inThis either empty, or a vertex, or a common edge or face and h is a measure of the mesh-size, namely, h = max_T_∈T_hh_T.

We assume that the family of triangulations is regular, i.e., for any T ∈ Thand any h, the regularity condition (9) is satisfied with a uniform σ.

Associated with the triangulationThwe introduce the global space

RTk(Th) ={v ∈ H(div, Ω) : v|T ∈ RTk(T ) ∀T ∈ Th} (34) When no confusion arises we will drop theThfrom the definition and callRTkthe global space. A fundamental tool in the error analysis is the operator

Π_h: H(div, Ω)∩

T∈Th

H¹(T )ⁿ −→ RTk

defined by

Π_hv|T = Π_Tv ∀T ∈ Th.

We have to check that Πhv ∈ RTk. Since by definition ΠTv ∈ RTk(T ), it only remains to see that Πhv∈ H(div, Ω).

First we observe that a piecewise polynomial vector function is in H(div, Ω) if and only if it has continuous normal component across the elements (this can be verified by applying the divergence theorem). But, since v ∈ H(div, Ω), the continuity of the normal component of Πhv follows from (b) of Lemma 3.1 in view of the degrees of freedom (18) in the definition of ΠT.

The finite element space for the approximation of the scalar variable p is the standard space of, not necessarily continuous, piecewise polynomials of degree k, namely,

P_k^d(Th) ={q ∈ L²(Ω) : q|T ∈ Pk(T ) : ∀T ∈ Th} (35) where the d stands for “discontinuous”. Also in this case we will write onlyP_k^dwhen no confusion arises. Observe that, since no derivative of the scalar variable appears in the weak form, we do not require any continuity in the approximation space for this variable.

In the following lemma we give two fundamental properties for the error analysis.

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Lemma 3.5. The operator Π_hsatisfies

Ω

div(v− Πhv) q dx = 0 (36)

∀v ∈ H(div, Ω) ∩

T∈ThH¹(T )ⁿand∀q ∈ P_k^d. Moreover,

divRTk =P_k^d. (37)

Proof. Using (18) and (19) it follows that, for any v∈ H¹(T )ⁿand any q∈ Pk(T ),

T

div(v− ΠTv)q dx =−

T

(v− ΠTv)· ∇q dx +

∂T

(v− ΠTv)· n q = 0 thus, (36) holds.

It is easy to see that divRTk ⊂ P_k^d. In order to see the other inclusion recall that from Lemma 2.2 we know that div : H¹(Ω)ⁿ→ L²(Ω) is surjective. Therefore, given q∈ P_k^dthere exists v ∈ H¹(Ω)ⁿ such that div v = q. Then, it follows from (36) that div Π_hv = q and so (37) is proved.

Introducing the orthogonal L²-projection Ph: L²(Ω)→ P_k^d, properties (36) and (37) can be summarized in the following commutative diagram

H¹(Ω)^{n div}−→ L²(Ω) Πh

⏐⏐

⏐⏐Ph

RTk −→ Pdiv _k^d −→ 0

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where, to simplify notation, we have replaced H(div, Ω)∩

T∈ThH¹(T )ⁿby its subspace H¹(Ω)ⁿ.

Our next goal is to give error estimates for the mixed finite element approxima- tion of problem (13), namely, (uh, ph)∈ RTk× P_k^ddefined by

⎧⎪

⎪⎨

⎪⎪

⎩

Ω

µ uh· v dx −

Ω

phdiv v dx = 0 ∀v ∈ RTk

Ω

q div uhdx =

Ω

f q dx ∀q ∈ P_k^d.

(39)

It is important to remark that, although we are considering the particular case of the Raviart–Thomas spaces on simplicial elements, the error analysis only makes use of the fundamental commutative diagram property (38) and of the approximation properties of the projections Π_hand P_h. Therefore, similar results can be obtained for other finite element spaces.

Lemma 3.6. If u and u_hare the solutions of (15) and (39) then,

u − uhL²(Ω)≤ (1 + aL^∞(Ω)µL^∞(Ω))uh− ΠhuL²(Ω).

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Proof. Subtracting (39) from (15) we obtain the error equations

Ω

µ (u− uh)· v dx −

Ω

(p− ph) div v dx = 0 ∀v ∈ RTk (40)

and,

Ω

q div(u− uh) dx = 0 ∀q ∈ P_k^d. (41) Using (36) and (41) we obtain

Ω

q div(Πhu− uh) dx = 0 ∀q ∈ P_k^d

and, since (37) holds, we can take q = div(Πhu− uh) to conclude that div(Π_hu− uh) = 0.

Therefore, taking v = Πhu− uhin (40) we obtain

Ω

µ (u− uh)· (Πhu− uh) dx = 0 and so,

Πhu− uh²_L²_(Ω)≤ aL^∞(Ω)

Ω

µ (Πhu− u)(Πhu− uh) dx

≤ aL^∞(Ω)µL^∞(Ω)Πhu− uL²(Ω)Πhu− uhL²(Ω)

and we conclude the proof by using the triangle inequality.

As a consequence, we have the following optimal order error estimate for the approximation of the vector variable u.

Theorem 3.2. If the solution u of problem (14) belongs to H^m(Ω)ⁿ, 1≤ m ≤ k+1, there exists a constant C depending onaL^∞(Ω),µL^∞(Ω), k, n and the regularity constant σ, such that

u − uhL²(Ω)≤ Ch^m∇^muL²(Ω).

Proof. The result is an immediate consequence of Lemma 3.6 and Theorem 3.1.

In the next theorem we obtain error estimates for the scalar variable p. We will use that

v − ΠhvL²(Ω)≤ ChvH¹(Ω) ∀v ∈ H¹(Ω) (42) which follows from a particular case of Theorem 3.1. In particular,

ΠhvL²(Ω)≤ CvH¹(Ω). (43)