Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 142

see §5. Here the given differential equation of order 2 is split into a system of
equations of first order. The mixed method induces a softening effect that is desired
in some difficult problems of solid mechanics; cf. Ch. VI, §3. There is a different
reason for a split of equations of fourth order into two equations of second order as
in Problem 4.24 or with Kirchhoff plates in Ch. VI, §5. The mixed method admits
the use of C 0 elements while conforming methods with the standard variational
formulation require C 1 elements.
3. Modeling boundary conditions. In some cases it is more convenient to have
a boundary condition u | − g =

0 as an explicit constraint than to incorporate
the boundary values into the finite element functions. Here the Lagrange multi-
plier models ∂u/∂n or, more generally, the multiple that is encountered in natural
boundary conditions. Similarly, the C 0 continuity is a handicap in domain decom-
position methods and is replaced by explicit matching conditions. This holds in
particular for mortar elements ; see Bernardy, Maday, and Patera [1994] or Braess,
Dahmen, and Wieners [2000].
4. Mixed elements that are equivalent to nonconforming methods. Often one
finds mixed methods that are equivalent to nonconforming elements. While the
nonconforming elements are more easily implemented, the mixed method may
admit an easier proof of convergence; see the DKT element for Kirchhoff plates
in Ch. VI, §5 and the connection between the nonconforming P 1 element and the
Raviart-Thomas element described by Marini [1985].
5. Saddle point problems with penalty terms. Problems with a large parameter
are often rewritten as a saddle point problem with a small penalty term. Examples
are the flow of a nearly incompressible fluid and the Reissner-Mindlin plate; see
Problem 4.19, Ch. VI, §§3 and 6.
6. A posteriori error estimates via saddle point problems. Nearly optimal
solutions of saddle point problems provide lower estimates of the (minimal) value
of variational problems and thus also a posteriori error estimates; see §9.

Problems

4.16 Show that the inf-sup condition (4.8) is equivalent to the following decom-
position property: For every u ∈ X there exists a decomposition

u = v + w

with v ∈ V and w ∈ V ⊥ such that

w X ≤ β − 1 Bu M ,

where β> 0 is a constant independent of u .