Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 144

4.21 The pure Neumann Problem (II.3.8)
− u = f

in ,

∂u
∂ν = g

on ∂

is only solvable if fdx + gds = 0. This compatibility condition follows

by applying Gauss' integral theorem to the vector field

∇ u . Since u + const is a

solution whenever u is, we can enforce the constraint

udx = 0 .

Formulate the associated saddle point problem, and use the trace theorem and
the second Poincare inequality to show that the hypotheses of Theorem 4.3 are
satisfied.

4.22 Let a, b , and c be positive numbers. Show that a ≤ b + c implies a ≤
b 2 /a + 2 c .

4.23 Show the equivalence of the conditions (4.24) and (4.25). For the nontrivial
direction, use the same argument as in the derivation of (4.26); cf. Braess [1996].

4.24 Let u be a (classical) solution of the biharmonic equation
2 u = f in ,

∂u
∂ν =

u = 0 n ∂.

Show that u ∈ H 0 together with w ∈ H 1 is a solution of the saddle point problem

0 for all η ∈ H 1 ,
( ∇ w, ∇ v) 0 , = (f, v) 0 , for all v ∈ H 0 .
Suitable elements and analytic methods can be found, e.g., in Ciarlet [1978] and
in Babuska, Osborn, and Pitkaranta [1980].

(w, η) 0 , + ( ∇ η, ∇ u) 0 , =

4.25 Equations of the form
a(u, v) + b(v, λ) = f, v for all v ∈ X,

b(u, µ) + c(σ, µ) = g, µ for all µ ∈ M, ( 4 . 28 )

c(τ, λ) + d(σ,τ) = h, τ for all τ ∈ Y

are sometimes called double saddle point problems . Rearrange (4.28) to obtain a
standard saddle point problem.