Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 105

Chapter III

Nonconforming and Other Methods

In the theory of conforming finite elements it is assumed that the finite element
spaces lie in the function space in which the variational problem is posed. More-
over, we also require that the given bilinear form a( · , · ) can be computed exactly
on the finite element spaces. However, these conditions are too restrictive for many
real-life problems.
1. In general, homogeneous boundary conditions cannot be satisfied exactly for
curved boundaries.
2. When we have variable coefficients or curved boundaries, we can only com-
pute approximations to the integrals needed to assemble the stiffness matrix.
3. For plate problems and in general for fourth order elliptic differential equa-
tions, conforming methods require C 1

elements, and this leads to very large

systems of equations.
4. We may want to enforce constraints only in the weak sense. A typical example
is the Stokes problem, where the variational problem is posed in the space of
divergence-free flows,
{ v ∈ H 0 () n

0 for all λ ∈ L 2 () } .
The constraint leads to saddle point problems, and we can only take into
account finitely many of the infinitely many constraints.
In this chapter we show that these types of deviations from the theory of
conforming elements are admissible and do not spoil convergence. In admitting
them, we are committing what are called variational crimes .
In §1 we establish generalizations of Cea's lemma, and examine its use by
looking at two applications. Then we give a short description of isoparametric ele-
ments. §§3 and 4 contain deep functional analytic methods which are of particular
importance for the mixed methods of mechanics. We illustrate them in §§6 and
7 for the Stokes problem. §5 prepares the reader for nonstandard applications of
saddle point problems.
§§8 and 9 will be concerned with a posteriori error estimates for finite element
solutions. Here arguments from the theory of nonconforming elements and mixed
methods enter even if we deal with conforming elements.
We should mention that the theory described in §3 has also recently been used
to establish the convergence of difference methods and finite volume methods.

; ( div v, λ) 0 , =