Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 75

5.15 Show that for a triangulation of a simply connected domain, the number of
triangles plus the number of nodes minus the number of edges is always 1. Why
doesn't this hold for multiply connected domains?

5.16 When considering the cubic Hermite triangle, there are three degrees of free-
dom per vertex and one per triangle. By the results of the previous problem, we know
that the dimension must be smaller than for the standard Lagrange representation.
Where are the missing degrees of freedom?

5.17 Let f ∈ L 2 () , and suppose u and u h are the solutions of the Poisson equation
− u = f in H 0 () and in a finite element space S h ⊂ C 0 () , respectively. By
construction ∇ u and ∇ u h are at least L 2 functions. By the remark in Example 2.10,
we know that the divergence of ∇ u is an L 2 function. With the help of Problem 5.14,

show that this no longer holds for the divergence of ∇ u h in general.

5.18 Show that the piecewise cubic continuous quadrilateral elements whose
restrictions to the edges are quadratic polynomials, are exactly the serendipity class
of eight node elements.
Hint: First consider a rectangle with sides parallel to the axes.

5.19 To construct triangular elements based on quadratic polynomials, consider
the subspace of functions whose normal derivatives on the three edges are constant.
Find the dimension of this space, distinguishing the cases when it is a right triangle
or not.

5.20 The set of cubic polynomials whose restrictions to the edges of a triangle are
quadratic form a 7-dimensional space. Give a basis for it on the unit triangle (5.9).
— We will later encounter the cubic bubble function B 3 . The result of this problem
can be identified with P 2 ⊕ B 3 .