This section will be mainly devoted to the approximation of Hi(O) and its subspaces of the form
HJ
D(O). We shall however sketch the results concerning the approximation of H2'(O). Standard approximations of Sobolev spaces can be subdivided into two classes: conforming and nonconforming methods. Even though nonconforming methods will be studied in the context of hybrid finite element methods, their importance makes it useful to introduce them here. We refer to CIARLET [A], BABUSKA-AZIZ [A] or RAVIART-THOMAS [D] for a detailed presentation of the following results.111.2.1 Conforming methods
Conforming methods are the most natural of finite element methods. They yield internal approximations in the sense that they enable us to build finite dimensional subspaces of the function space that we want to approximate.
Given a partition of the domain 0, into triangles or quadrilaterals, a con-forming approximation of Hi (0) is a space of continuous functions defined by a finite number of parameters (or degrees of freedom).
The last condition is usually met by using a space of piecewise polynomial functions or functions obtained from polynomials by a change of variables like (using the notations of Section III.1)
(2.1) Vh K
I
= v ' 0 F-i ,where K = F(
K)
and ii is a polynomial function onK.
Continuity is obtained by a clever choice of degrees of freedom.Remark 2.1: For triangular elements it is usual and convenient to use piecewise polynomial functions on K. For quadrilaterals it is essential to use (2.1). It must then be noted that vhlK is not in general a polynomial on K . This will be the case only for affine transformations. 0
To give a more precise definition of our fin ite element approximations we shall need a few definitions. Let us define on an element J{
(2.2) Pk (K) : the space of polynomials of degree ::; k.
100 Function Spaces and Finite Element Approximations §III.2 The dimension of Pk(K) is ~(k
+
l)(k+
2) for n = 2, and for n = 3, it is i(k+
l)(k+
2)(k+
3). It will sometimes be convenient to define (for n = 2) (2.3) Pk1 ,k,(K)=
{P(X!,X2)I
p(XI,X2)= L
aijxi xH
i<k1
jS,k,
the space of polynomials of degree ~ kl in Xl and ~ k2 in X2 . In the same way we can define Pk 1,k"k3(K) for n
=
3. The dimensions of these spaces are respectively (kl+
1)(k2+
1) and (kl+
1)(k2+
1)(k3+
1). We then define(2.4) () { Pk ,k(K) for n = 2
Qk K =
Pk ,k,k,(K) for n
=
3.We shall also need polynomial spaces on the edges (or faces) of the ele-ments. Using the notations of Section III.1.2, we define
Functions of Rk(8K) are polynomials of degree ~ k on each side (or face) of K . They do not have to be continuous at vertices (or edges). The dimensions of Rk(8K) and Tk(8K) are respectively for k
2:
1:- 3( k
+
1) and 3k for triangles, - 4( k+
1) and 4k for quadrilaterals,- 2(k
+
l)(k+
2) and 2(k2+
1) for tetrahedra.For hexahedra, it will usually be more convenient to consider functions in Qk( ei) in the definition of Rk(8K) and Tk(8K).
To define a finite element, we must, following CIARLET [A), specify three things.
- The geometry: we choose a reference element i< and a change of variables F(x), and we set K = F(i<).
- A set
P
of polynomials on i<. ForpEP
we define, on K, p= po
F-I.- A set of degrees of freedom
E,
that is, a set of linear forms {idl~i~dim P onP.
We say that this set is unisolvent when these linear forms are linearly independent, i.e., the knowledge ofi ;(p)
for all i completely definesp.
A finite element is of Lagrange type if its degrees of freedom are point values, that is, one is given a set
{ad
I ~ i ~ dim P of points in i< and one defines(2.7) 1 ~ i ~ dim
P.
§II1.2 Mixed and Hybrid Finite Element Methods 101 For the approximation of Hi (0), Lagrange type elements will be sufficient but approximating H2(0) requires Hermite type elements, that is, degrees of freedom involving derivatives.
Remark 2.2: The reader should be aware that not any choice of points will yield an unisolvent set of degrees of freedom. Moreover the points have to be chosen in order to ensure interelement continuity. 0
Example 2.1: Affine finite elements
This is the most classical family of finite elements. The reference element is the triangle
K
of Figure III.1 and we use the affine transformation(2.8)
F(x) =
Xo+ Bx.
The element J{ is still a triangle and it is not degenerate provided det B :/: O.
We now take P
=
Pk (K) and choose an appropriate set of degrees of freedom.The standard choice for k ::; 3 are presented in Figure III.2.
"
"z
(0,1)F (1,0)
Figure lII.l
P2(K)
Figure 111.2: Standard conforming elements
One notes that this choice of points enSures continuity at interfaces. 0
102 Function Spaces and Finite Element Approximations §1I1.2 Example 2.2: /soparametric triangular elements
We use the same reference element and the same set
P
as in the previous example. We now take the transformationF(
x) such that each of its components Fi belongs to Pk(K). For k = 1 nothing is changed but for k ~ 2, the element K now has curved boundaries. We present the case k=
2 in Figure 111.3.a,
Figure 111.3: Isoparametric triangle of degree two
Using such curved triangles enables us to obtain a better approximation of curved boundaries. It must be noted that the curvature of boundaries introduces addi-tional terms in the approximation error and the curved elements should be used only when they are really necessary (CIARLET-RAVIART [C] or CIARLET [A]).
Example 2.3: /soparametric quadrilateral elements
This is also a very classical family of finite elements. The reference element is the square K =]0, l[x]O,
1[.
We takeP
= Qk(K) and a transformation F with each component in Qk(K). We present the standard choice of degrees of freedom for k ::; 2 in Figure IlI.4. It must be noted that we need FE (Ql(K))2 to define a general straight-sided quadrilateral.ii,
---,.
ii,a) The Q 1 isoparametric element Figure 111.4
§II1.2
R
1
Mixed and Hybrid Finite Element Methods
,
3 a~Finally we recall that it is possible to eliminate internal nodes to get the so-called serendipity finite elements. For instance if we take
4 8
Approximating
H2(D.)
will require continuity of derivatives at interelement boundaries and leads to the introduction of elements in which values of the derivatives are used as degrees of freedom. The simplest Hermite type element is the P3 triangle of Figure III.6a.104 Function Spaces and Finite Element Approximations
a) P3-triangle b) Argyris' triangle
• value of the function
®
values of the function and its first derivative@
values of the function and its first and second derivative value of the normal derivativeFigure 111.6
§III.2
Here the degrees of freedom are values of the function and its derivative at vertices plus a point value at barycenter. This element does not enable us to build an approximation of H2(0,). To do so, one must use Argyris' triangle (Figure III.6b) where polynomials of degree 5 are used. (Composite elements may also be used.) For quadrilaterals the analogues are easily built. The difficulty of building approximations of H2(0,) by standard methods was one of the major reason for the introduction of various kinds of mixed or hybrid methods for plate problems (cf. Section IY.5 or Section VII.1).
We now have to say a few words about the approximation of a given function v by the finite element spaces just described or similar ones. We shall not give proofs, for which we refer to CIARLET [A], STRANG-FIX [A], CLEMENT [A].
We must first define the interpolate of v. For a general set of degrees of freedom
{fd
onk,
we define T h V by(2.9) 1 ::;
i::;
dimP.
The operator M must be a well-defined continuous form. When the linear forms fi are defined by (2.5), it is natural to set
(2.10)
This definition makes sense only when
v
is a continuous function which is not the case when v E H1(0,). For Lagrange type elements in JR2 or IR?,v
E H2( k) is a sufficient condition for (2.10) to be justified and Th V is just the Lagrange interpolate, in the classical sense, ofv.
For v E H1(0,), CLEMENT [A] has defined a continuous interpolate Th using averages of u instead of point values. This also implies a more elaborate use of reference elements. In particular, the operator Th V is no longer defined on an element. In fact the nodal§III.2 Mixed and Hybrid Finite Element Methods 105 values of
rh
ii depend on the value of ii on the adjacent elements through an averaging process.Once
rh
ii is defined, we can define on K, (2.11)We rapidly recall a few classical results. We refer the reader to CIARLET [A] for a detailed presentation. We first consider the case of affine elements, assuming first rh to be defined by the usual interpolate (2.10).
Proposition 2.1: If the mapping
F
is affine, that isF(x)
= Xo+ Bx,
and if rhPk=
Pk for any Pk E Pk(K), we have for v E H'(0.), m ::; s, 1<
s ::;k+
1(2.12)
The proof uses (1.43), its reciprocal (1.44) and the classical results stated below. 0
Lemma 2.1: 1·lk+l,n is a norm on Hk+l(0.)j Pk(0.), equivalent to the standard quotient norm. 0
From this one deduces another classical result:
Lemma 2.2: (Bramble-Hilbert lemma) Let L be a continuous linear form on Hk+1(0.) such that L(Pk) = 0 for any Pk E Pk(0.). Then there exists a constant c (depending on Land 0.) such that one has
(2.13) IL(v)1 ::; c Ivlk+l,n. 0
Results similar to (2.12), although more complex, can be obtained for general isoparametric elements (CIARLET [A], CIARLET-RAVIART [A,CD.
Let then hK be the diameter of K. Provided some classical conditions on the shape of elements forbidding degeneracy (CIARLET [AD are fulfilled, relation (2.12) can be converted into a relation involving a power of hK . For affine elements one defines for instance
(2.14)
where P K is the diameter of the largest inscribed disk (or sphere) in K.
We shall in the following always assume that the interpolation operator rh is defined by the method of CLEMENT [A], that is, by a local projection instead
106 Function Spaces and Finite Element Approximations §1II.2 may recall however that (2.16) can be written as a condition on angles excluding degenerate elements. For general curved elements there is also a condition on
For more general partitions including general isoparametric elements, the result is qualitatively the same: we have an O( hk) approximation provided the family of partitions is regular in a sense to be precised.
We also refer the reader to JAMET [A] where some degenerate cases are analyzed.
From the elements described above, we can build approximations of
Hl(O)
andH2(O).
The idea is, of course, to use functions whose restriction to an element belongs to a set of polynomial (or image S of polynomial) functions.Let Sk(K) be a subspace of Pk(K). We define for a partition Th of 0 (2.18)
§II1.2 Mixed and Hybrid Finite Element Methods Remark 2.3: In the two-dimensional case, for s
=
1 and s'c'(Sk, Th ) C C·-1(!:1) . although, this is not true for H·(0.) 0
107 2, we have We shall reduce this notation when no confusion is to be feared and write (2.19)
when Th is built from triangles, and Sk = Pk (= the space of polynomials of degree ~ k). In the same way we shall write
(2.20)
when Th is built from quadrilaterals.
We shall often use in our constructions bubble functions. For an element
I<
a bubble function is a function vanishing onoI<.
Thus we say that Sk is a set of bubble functions if Sk CH{j(I<) .
We then denote(2.21)
and we shall use the compact notation
(2.22) { Bk = B(Pk
n H6(I<)),
B[kJ=
B(Qkn H6 (I<)) ,
when no ambiguity will be possible.Spaces of bubble functions will be used to build enriched spaces. For instance the space ,c~ $ B3 will be useful in Chapter VI for the approximation of Stokes problem.
When approximating a standard elliptic problem, the finite element spaces introduced up to now can be used directly in the variational fomulation of the problem and error estimates follow from interpolation error estimates (CIARLET [AD. In many cases, however, nonconforming methods have proved to yield accurate (and sometimes easier to handle) approximations.
111.2.2 Nonconforming methods
We shall meet later nonconforming methods when studying hybrid finite element methods. In many cases, it will however be more convenient to see them in the frame of external approximations, which we now define.
Let us consider a variational problem (with
f
E V'), (2.23)a(u,v) = (J,v)v'xv,
Vv E V, u E V,where V is some Hilbert space and
a(u,
v) a bilinear (coercive) form on V xV.108 Function Spaces and Finite Element Approximations §II1.2 Suppose we can find a larger space S J V, such that there exists a canonical extension
ae .)
to S x S, satisfyingUsing standard coerciveness and continuity assumptions, one gets from (2.23) and (2.26) a result known as Strang's lemma (CIARLET [A], STRANG-FIX
The last term can be seen as a consistency term: it measures how well the exact solution satisfies the discrete equation. This term vanishes when Vh C V and such that error estimates obtained from (2.27) be "optimal". Optimality is here relative to the degree of the polynomials from which the approximation is built: