Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 83

Inverse Estimates

The above approximation theorems have the form

u − Iu m,h ≤ ch t − m

u t ,

where m is smaller than t . For the moment we ignore the fact that on the right-hand
side, the norm |·| t . Thus, the approximation

error is measured in a coarser norm than the given function. In a so-called inverse
estimate , the reverse happens. The finer norm of the finite element functions will
be estimated by a coarser one (obviously, this does not work for all functions in
the Sobolev space).

· t may be replaced by the semi-norm

6.8 Inverse Estimates. Let (S h ) be an affine family of finite elements consisting of
piecewise polynomials of degree k associated with uniform partitions. Then there
exists a constant c = c(κ, k, t) such that for all 0 ≤ m ≤ t,

v h t,h ≤ ch m − t

v h m,h for all v h ∈ S h .

Sketch of the proof . We can reduce the proof to the discussion of a reference element
by using the transformation formula 6.6. It suffices to show that

| v | t,T ref ≤ c | v | m,T ref for v ∈ ref ( 6 . 12 )

with c = c( ref ) . The extension to elements of size h proceeds just as in the proof
of Theorem 6.4. This leads to the factor ch m − t in the estimate. Then summing the
squares of the expressions over all triangles or quadrilaterals leads to the desired
assertion.
To establish (6.12), we make use of the fact that the norms · m,T ref

are equivalent on the finite dimensional space ref ⊕ P m − 1 . Let Iv ∈ P m − 1 be a
polynomial that interpolates v at fixed points. Since t>mi 1, we have

· t,T ref and

| Iv | t = 0.

Combining these facts, we obtain from the Bramble-Hilbert lemma

| v | t =| v − Iv | t ≤ v − Iv t
≤ c v − Iv m
= c | v | m ,

and (6.12) is proved.
In the Approximation Theorem 6.4 and the Inverse Estimate 6.8, the exponents
in the term with h correspond to the difference between the orders of the Sobolev
norms. This has been established by moving back and forth to and from the reference
triangle. This technique is called a scaling argument .