Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 89

§ 7. Error Bounds for Elliptic Problems
of Second Order

Now we are ready to establish error estimates for finite element solutions. Usually
error bounds are derived first with respect to the energy norm. The extension to
the L 2 -norm is performed by a duality technique that is often found in proofs of
advanced results. We are looking for bounds of the form

u − u h ≤ ch p ( 7 . 1 )

for the difference between the true solution u and the approximate solution u h in S h .
Here p is called the order of approximation . In general, it depends on the regularity
of the solution, the degree of the polynomials in the finite elements, and the Sobolev
norm in which the error is measured.

Remarks on Regularity

1, H 0 () ⊂ V ⊂ H m () , and suppose a( · , · ) is a
V -elliptic bilinear form. Then the variational problem

7.1 Definition. Let m ≥

a(u, v) = (f, v) 0 for all v ∈ V

is called H s -regular provided that there exists a constant c = c(, a, s) such that
for every f ∈ H s − 2 m () , there is a solution u ∈ H s () with

u s ≤ c f s − 2 m . ( 7 . 2 )

2 m . We will
drop this restriction later, in Ch. III, after norms with negative index are defined.
Regularity results for the Dirichlet problem of second order with zero boundary
conditions can be found, e.g., in Gilbarg and Trudinger [1983] and Kadlec [1964].
For simplicity, we do not present the most general results; see the remarks for
Example 2.10 and Problem 7.12.

In this section we will make use of this definition only for s ≥

7.2 Regularity Theorem. Let a be an H 0 -elliptic bilinear form with sufficiently
smooth coefficient functions.
(1) If is convex, then the Dirichlet problem is H 2 -regular.
(2) If has a C s 2 , then the Dirichlet problem is H s -regular.

boundary with s ≥