Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 153

The approximating functions for σ h no longer involve continuity conditions,
and each basis function has support on a single triangle. If we eliminate the asso-
ciated variables by static condensation, the resulting equations are just as sparse as
before the elimination process. In addition, we have avoided the costly construction
of a basis of Raviart-Thomas elements.
A further advantage is that the Lagrange multiplier can be regarded as a finite
element approximation of u on the edges. Arnold and Brezzi [1985] used them to
improve the finite element solution.

Mesh-Dependent Norms for the Raviart-Thomas Element

Finite element computations with the Raviart-Thomas elements may also be
analyzed in the framework of primal mixed methods, i.e., with the pairing
H 1 () , L 2 () . Since the tangential components of the functions in (5.8) may
have jumps on inter-element boundaries, in this context the elements are non-
conforming and we need mesh-dependent norms which contain edge terms in
adddition to broken norms

* -

1 / 2

+ h

, τ

2
0

2
0 ,e

/

τ 0 ,h : =

e ⊂ h τn

,

( 5 . 14 )

*
,
T

-

1 / 2

+ h − 1
e

2
1 ,T

2
0 ,e

/

| v | 1 ,h : =

∈ T h | v | h J(v)

.

⊂

Here, h : =∪ T (∂T ∩ ) is the set of inter-element boundaries. On the edges of

h the jump J(v) of v and the normal component τn of τ are well defined. We
note that both τn and J(v) change sign if the orientation of an edge is reversed.
Therefore, the product is independent of the orientation.
The continuity of the bilinear form a( · , · ) is obvious. Its coercivity follows

from

τ 0 ,h ≤ C τ 0 for all τ ∈ RT k

which in turn is obtained by a standard scaling argument. The bilinear form b( · , · )
is rewritten by the use of Green's formula

b(τ, v) =− τ · grad vdx + J(v)τnds. ( 5 . 15 )

T

T ∈ T

h

Now its continuity with respect to the norms (5.14) is immediate.