Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 179

L − 1 1 instead of − 1 , where

L − 1 is an approximate inverse. Such an approximate inverse is implicitly used
with hierarchical estimators. Although (8.5) and (8.6) define also approximate
inverses, these (local) approximate inverses provide estimators that are equivalent
to residual estimators; see Verf urth [1996]. Thus it is not clear whether they are
more efficient than residual estimators in the case of large condition numbers.
Babuska, Duran, and Rodrıguez [1992] show that the efficiency of estimators
can be much worse for unstructured grids than for regular ones.

In this case it is assumed to be better to compute

Local Mesh Refinement and Convergence

In finite element computations using local grid refinement, we generally start with a
coarse grid and continue to refine it successively until the estimator η T,R is smaller
than a prescribed bound for all elements T . In particular, those elements where
the estimators give large values are the ones which are refined. The geometrical
aspects have already been discussed in Ch. II, §8.
This leads to a triangulation for which the estimators have approximately
equal values in all triangles. Numerical results obtained using this simple idea are
quite good.
The above approach can be justified heuristically. Suppose the domain in
d -space is divided into m (equally large) subdomains where the derivatives of the
solution have different size. Suppose an element with mesh size h i in the i -th
subdomain contributes c i h i to the error, where α>d . The subdomains involve
different factors c i , but are all associated with the same exponent α .Ifthe i -th
subdomain is divided into n i parts, then h i = n 1 /d , and the total error is of order

i

c i h α − d n i c i h i =

. ( 8 . 32 )

i

i i

Our aim is to minimize the expression (8.32) subject to i n i = i h − d = const.

i

The optimum is a stationary point of the Lagrange function

+ λ
i

const .

c i h α − d h − d

L (h, λ) : = − ( 8 . 33 )

i i

i

If we relax the requirement that the n i be integers, then we can find the optimum
by differentiating (8.33), which leads to c i h i =

dλ
α d . This is just the condition −

that the contributions of all elements be equal.

The convergence of the finite element computations with the refinement strat-
egy above is not obvious. A first proof was established by D orfler [1996], and it
was extended later by Morin, Nochetto, and Siebert [2002]. The general scheme