TABLE OF CONTENTS 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. Moreover, 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.
In general, homogeneous boundary conditions cannot be satisfied exactly for curved boundaries.
When we have variable coefficients or curved boundaries, we can only compute approximations to the integrals needed to assemble the stiffness matrix.
For plate problems and in general for fourth order elliptic differential equations, conforming methods require C 1 elements, and this leads to very large systems of equations.
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,
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 C a s lemma, and examine its use by looking at two applications. Then we give a short description of isoparametric elements. 3 and 4 contain deep functional analytic methods which are of particular importance for...
