Numeerisen analyysin ja laskennallisen
tieteen seminaari
27.3.2006 klo
14.15
U356
Antti Hannukainen, TKK, Matematiikan laitos
A posteriori error
estimation for elliptic BVP's
All computational processes normally involve computational errors of
various types. No matter how appropriate the underlying mathematical
model is, these errors may be excessively large and render the solution
practically useless. In truly reliable computations this is avoided
with a posteriori error estimation of the obtained numerical solution.
However, no unified a posteriori error estimation approach exists and
all techniques are strongly connected to the underlying problem or
computational method.
Elliptic problems are usually used as a testbench for different error
control techinques. In this context the error can be measured in
several different ways. Error in the energy norm is used to measure the
overall quality of the solution. In some cases this is not sufficent
and more sophisticated error measures are required. For example, error
over some critical part of the solution domain can be expressed in
terms of linear functionals.
In this talk we present two complementing a posteriori error estimation
techniques for elliptic BVP. Corresponding estimates are derived for
error measured both in the energy norm and in terms of linear
functionals. First technique is based on superconvergence properties of
averaged gradients and gives indication of the error source. However,
superconvergence requires the solution to be a FEM-solution, so this
method can be applied only in the context of FEM. The second technique
provides guaranteed two-sided bounds for the error and is applicable
for a wide class of approximations independently of the methods used to
compute them. The bounds can be made as close to the true error as
computational resources allow. Both estimates are easy to code and all
needed constants are computable and independent of the mesh. Comparison
and benefits of the proposed methods are demonstrated in several
numerical tests.