ubc
Adaptivity in anisotropic finite element calculations
2006-05-09
[Electronic ed.]
prv
Universitätsbibliothek Chemnitz
Universitätsbibliothek Chemnitz, Chemnitz
Fakultät für Mathematik
male
Rostow am Don
male
male
male
male
When the finite element method is used to solve boundary value problems, the
corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters.
There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation.
Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment.
In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown.
Numerical experiments for the equilibrated residual method, for the hierarchical
error estimator and for the adaptive algorithm confirm the theory. The adaptive
algorithm shows its potential by creating the anisotropic mesh for the problem
with the boundary layer starting with a very coarse isotropic mesh.
510
Anisotropes Gitter
Finite-Elemente-Methode
Konvergenz
adaptive algorithm
anisotropic a posteriori error estimation
anisotropic mesh
equilibrated residual method
finite elements
hierarchical error estimator
singularly perturbed reaction-diffusion equation
triangular mesh
urn:nbn:de:swb:ch1-200600815
Technische Universität Chemnitz
dgg
Technische Universität Chemnitz, Chemnitz
Sergey
Grosman
M. Sc.
1978-08-15
aut
Thomas
Apel
Prof. Dr.
dgs
rev
Bernd
Heinrich
Prof. Dr.
rev
Gert
Lube
Prof. Dr.
rev
eng
2006-02-02
2006-04-21
born digital
doctoral_thesis