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Mixed Finite Element Methods

Ricardo G. Duran

1 Introduction

Finite element methods in which two spaces are used to approximate two dif-ferent variables receive the general denomination of mixed methods. In somecases, the second variable is introduced in the formulation of the problembecause of its physical interest and it is usually related with some derivativesof the original variable. This is the case, for example, in the elasticity equa-tions, where the stress can be introduced to be approximated at the sametime as the displacement. In other cases there are two natural independentvariables and so, the mixed formulation is the natural one. This is the caseof the Stokes equations, where the two variables are the velocity and thepressure.

The mathematical analysis and applications of mixed finite element meth-ods have been widely developed since the seventies. A general analysis forthis kind of methods was first developed by Brezzi [13]. We also have tomention the papers by Babuska [9] and by Crouzeix and Raviart [22] which,although for particular problems, introduced some of the fundamental ideasfor the analysis of mixed methods. We also refer the reader to [32, 31], wheregeneral results were obtained, and to the books [17, 45, 37].

The rest of this work is organized as follows: in Section 2 we review somebasic tools for the analysis of finite element methods. Section 3 deals withthe mixed formulation of second order elliptic problems and their finite ele-ment approximation. We introduce the Raviart-Thomas spaces [44, 49, 41]and their generalization to higher dimensions, prove some of their basicproperties, and construct the Raviart-Thomas interpolation operator whichis a basic tool for the analysis of mixed methods. Then, we prove optimalorder error estimates and a superconvergence result for the scalar variable.

Departamento de Matematica, Facultad de Ciencias Exactas, Universidad de BuenosAires, 1428 Buenos Aires, Argentina. E-mail: rduran@dm.uba.ar

1

We follow the ideas developed in several papers (see for example [24, 16]).Although for simplicity we consider the Raviart-Thomas spaces, the erroranalysis depends only on some basic properties of the spaces and the inter-polation operator, and therefore, analogous results hold for approximationsobtained with other finite element spaces. We end the section recallingother known families of spaces and giving some references. In Section 4we introduce an a posteriori error estimator and prove its equivalence withan appropriate norm of the error up to higher order terms. For simplicity,we present the a posteriori error analysis only in the two dimensional case.Finally, in Section 5, we introduce the general abstract setting for mixedformulations and prove general existence and approximation results.

2 Preliminary results

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

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

Hk() = { L2() : D L2() || k},where

= (1, , n) , || = 1 + + n and D = ||

x11 xnnand the derivatives are taken in the distributional or weak sense.

Hk() is a Hilbert space with the norm given by

2Hk() =

||kD2L2().

Given Hk() and j IN such 1 j k we define j by|j|2 =

||=j|D|2.

Analogous notations will be used for vector fields, i.e., if v = (v1, , vn)then Dv = (Dv1, , Dvn) and

v2Hk() =n

i=1

vi2Hk() and |jv|2 =n

i=1

|jvi|2.

2

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

H10 () = { H1() : | = 0},

H1() = { H1() :

dx = 0}.

Also, we will use the standard notation Pk for the space of polynomialsof degree less than or equal to k and, if x IRn and is a multi-index, wewill set x = x11 xnn .

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

Given a function in a Sobolev space of a domain it is important toknow whether it can be restricted to , and conversely, when can a functiondefined on be extended to in such a way that it belongs to the originalSobolev space. We will use the following trace theorem. We refer the readerfor example to [38, 33] for the proof of this theorem and for the definitionof the fractional-order Sobolev space H

12 ().

Theorem 2.1 Given H1(), where IRn is a Lipschitz domain,there exists a constant C depending only on such that

H

12 ()

CH1().

In particular,L2() CH1(). (2.1)

Moreover, if g H 12 (), there exists H1() such that | = g andH1() CgH 12 ().

One of the most important results in the analysis of variational methodsfor elliptic problems is the Friedrichs-Poincare inequality for functions withvanishing mean average, that we state below (see for example [36] for thecase of Lipschitz domains and [43] for another proof in the case of convexdomains). Assume that is a Lipschitz domain. Then, there exists aconstant C depending only on the domain such that for any f H1(),

fL2() CfL2(). (2.2)The Friedrichs-Poincare inequality can be seen as a particular case of

the next result on polynomial approximation which is basic in the analysisof finite element methods.

3

Several different arguments have been given for the proof of the nextlemma. See for example [12, 25, 26, 51]. Here we give a nice argumentwhich, to our knowledge, is due to M. Dobrowolski for the lowest ordercase 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 withrespect to a subset of positive measure and any degree of approximationis an immediate extension of Dobrowolskis argument. For simplicity wepresent the proof for the L2-case (which is the case that we will use), but thereader can check that an analogous argument applies for Lp based Sobolevspaces (1 p < ).

Assume that is star-shaped with respect to a set B of positivemeasure. Given an integer k 0 and f Hk+1() we introduce the aver-aged Taylor polynomial approximation of f , Qk,Bf Pk defined by

Qkf(x) =1|B|

BTkf(y, x) dy

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

Tkf(y, x) =

||kDf(y)

(x y)!

.

Lemma 2.2 Let IRn be a domain with diameter d which is star-shapedwith respect to a set of positive measure B . Given an integer k 0 andf Hk+1(), there exists a constant C = C(k, n) such that, for 0 || k + 1,

D(f Qk,Bf)L2() C||1/2|B|1/2 d

k+1|| k+1fL2(). (2.3)

In particular, if is convex,

D(f Qk,f)L2() C dk+1|| k+1fL2(). (2.4)

Proof. By density we can assume that f C(). Then we can write

f(x) Tkf(y, x) = (k + 1)

||=k+1

(x y)!

10

Df(ty + (1 t)x) tk dt.

4

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

f(x)Qk,Bf(x) = k + 1|B|

||=k+1

B

10

(x y)!

Df(ty + (1 t)x) tk dt dy

and so,

|f(x)Qk,Bf(x)|2 dx C d

2(k+1)

|B|2

||=k+1

(

B

10|Df(ty+(1t)x)|tk dt dy

)2dx

C d2(k+1)

|B|2

||=k+1

(

B

10|Df(ty+(1t)x)|2 dt dy

)(

B

10

t2k dt dy)

dx.

Therefore,

|f(x)Qk,Bf(x)|2 dx C d

2(k+1)

|B|

||=k+1

B

10|Df(ty+(1t)x)|2 dt dy dx

(2.5)Now, for each ,

B

10|Df(ty + (1 t)x)|2dt dy dx

=

B

12

0|Df(ty+(1t)x)|2dt dy dx+

B

112

|Df(ty+(1t)x)|2dt dy dx =: I+II

Let us call g the extension by zero of Df to IRn. Then, by Fubinistheorem and two changes of variables we have

I

B

12

0

IRn|g(ty+(1t)x)|2 dx dt dy =

B

12

0

IRn|g((1t)x)|2 dx dt dy

=

B

12

0

IRn|g(z)|2(1 t)n dz dt dy 2n1|B|

|Df(z)|2 dz.

Analogously,

II

112

IRn|g(ty + (1 t)x)|2 dy dt dx =

112

IRn|g(ty)|2 dy dt dx

=

112

IRn|g(z)|2tn dz dt dx 2n1||

|Df(z)|2 dz.

5

Therefore, replacing these bounds in (2.5) we obtain (2.3) for = 0.On the other hand, an elementary computation shows that

DQk,Bf(x) = Qk||,B(Df)(x) || kand therefore, the estimate (2.3) for || > 0 follows from the case = 0applied to Df .

An important consequence of this result are the following error estimatesfor the L2-projection onto Pm.Corollary 2.3 Let IRn be a domain with diameter d which is star-shaped with respect to a set of positive measure B . Given an integerm 0, let P : L2() Pm be the L2-orthogonal projection. There exists aconstant C = C(j, n) such that, for 0 j m, if f Hj(), then

f PfL2() C||1/2|B|1/2 d

j |jf |L2().

Remark 2.1 Analogous results to Lemma 2.3 and its corollary hold forbounded Lipschitz domains because this kind of domains can be written as afinite union of star-shaped domains (see [25] for details).

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

Lemma 2.4 Let IRn be a bounded domain. Given f L2() thereexists v H1()n such that

divv = f in (2.6)

andvH1() CfL2() (2.7)

with a constant C depending only on .

Proof. Let B IRn be a ball containing and be the solution of theboundary problem {

= f in B = 0 on B

(2.8)

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

H2() CfL2()and therefore v = satisfies (2.6) and (2.7).

6

Remark 2.2 To treat Neumann boundary conditions we would need the ex-istence of a solution of divv = f satisfying (2.7) and the boundary conditionv n = 0 on .