Inexact Adaptive Finite Element Methods for Elliptic PDE Eigenvalue Problems [Elektronische Ressource] / Agnieszka Miedlar. Betreuer: Volker Mehrmann

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 12 Mo

Extrait

Inexact Adaptive Finite Element Methods for
Elliptic PDE Eigenvalue Problems
vorgelegt von
Magister Ingenieurin Agnieszka Międlar
aus Wrocław, Polen
Von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Martin Skutella
Gutachter: Prof. Dr. Volker Mehrmann
Gutachter: Prof. Dr. Carsten Carstensen (Humboldt-Universität zu Berlin)
zusätzliche
Gutachter: Prof. Ing. Zdeněk Strakoš, DrSc. (Charles University of Prague)
Tag der wissenschaftlichen Aussprache: 18.03.2011
Berlin 2011
D 83Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Hope for changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Content of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries 9
2.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Normed and inner product vector spaces . . . . . . . . . . . . . . . . 9
2.1.2 Partial differential operators . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Linear functionals and sesquilinear (bilinear) forms . . . . . . . . . . 11
2.1.4 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.5 Matrix theory and eigenvalue problems . . . . . . . . . . . . . . . . . 14
2.1.6 Backward error analysis and condition numbers . . . . . . . . . . . . 16
2.2 PDE eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Classical and variational formulation of elliptic eigenvalue problems . 17
2.2.2 The Galerkin method and the Finite Element Method (FEM) . . . . 18
2.2.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.4 The Adaptive Finite Element Method (AFEM) . . . . . . . . . . . . 25
2.3 The Generalized Algebraic Eigenvalue Problem . . . . . . . . . . . . . . . . 28
2.3.1 The Arnoldi/Lanczos method . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Homotopy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Perturbation results for the generalized eigenvalue problem . . . . . . 32
2.4 Continuous-discrete inner product and norm relations . . . . . . . . . . . . . 39
3 Model problems 43
3.1 A Laplace eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 A convection-diffusion eigenvalue problem . . . . . . . . . . . . . . . . . . . 45
4 Self-adjoint eigenvalue problem 47
4.1 A comparison of discretization and iteration errors . . . . . . . . . . . . . . . 47
4.1.1 A model problem and error estimates . . . . . . . . . . . . . . . . . . 48
4.1.2 Numerical examples - How exact the ’exact’ really is? . . . . . . . . . 50
4.2 AFEMLA - two way adaptation based on the iteration error . . . . . . . . . 51
ii4.2.1 Standard AFEM versus AFEMLA . . . . . . . . . . . . . . . . . . . . 52
4.2.2 The AFEMLA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Error estimates involving the algebraic error . . . . . . . . . . . . . . 60
4.2.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Functional perturbation results for PDE eigenvalue problems . . . . . . . . . 86
4.3.1 The functional backward error and condition number . . . . . . . . . 86
4.4 A combined a posteriori error estimator for self-adjoint eigenvalue problems . 97
4.4.1 A combined residual error estimator . . . . . . . . . . . . . . . . . . . 98
4.4.2 The balanced AFEM algorithm . . . . . . . . . . . . . . . . . . . . . 100
4.4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Non-self-adjoint eigenvalue problem 113
5.1 The Non-self-adjoint AFEMLA . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.1 The Non-self-adjoint AFEMLA algorithm . . . . . . . . . . . . . . . 114
5.1.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.3 Some error bounds for the eigenvalues and eigenfunctions . . . . . . . 130
5.2 An adaptive homotopy approach for non-self-adjoint eigenvalue problems . . 137
5.2.1 Homotopy method for an operator eigenvalue problem . . . . . . . . 138
5.2.2 A posteriori error estimates . . . . . . . . . . . . . . . . . . . . . . . 139
5.2.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Conclusions 167
7 Appendix 169
7.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Bibliography 172
iiiAcknowledgements
When eating a fruit, think of the person who planted the tree.
Vietnamese Proverb
Someone said that you need passion in your heart and oil in your head to write a Ph.D.
thesis. After this three years I realized these are the things which put you on the track, but
there is something else what keeps you on going and prevent from derailing on many turns
on the way. These are people around you, those who are there every single day, who believe
in you, often much stronger than you own do.
I would like to thank my advisor Prof. V. Mehrmann for taking a risk three years ago
and accepting me under his wings. For long hour discussions, answering questions even at
least appropriate moments and for showing that research can be a fascinating way of life.
Moreover, for continued involvement in making this group so special and supporting us not
only as great teacher but as an authority.
A very special thanks to my co-authors Prof. C. Carstensen who agreed to be myBMS men-
tor and J. Gedicke for our long debates, for introducing an adaptive world to me and provid-
ing theopenFFW [29] finite element framework for the numerical experiments. I would like
tothank Dr. M. Arioli, Prof. S. Friedland, Prof.J. Liesen, Prof. L.Grubišić, Prof.U. Hetma-
niuk,Prof.B.Parlett,Prof.R.Rannacher,Prof.R.Schneider,Prof.Z.Strakoš,Prof.L.N.Tr-
effethen, for inspiring discussions about mathematics and life. Special thanks to my former
advisor Prof. K. Ziętak who opened the world of the numerical methods in front of me.
Many thanks to my office-mates Lisa and Ann-Kristin, all present and previous colleagues
and my friends in Berlin and Wrocław for help, kind words which made this time nice,
preparing a cup of tee in the morning or simply for being there.
IappreciatetheBerlinMathematicalSchoolandtheDFGResearchCenterMatheon
for the financial support and the BMS One-Stop-Office team for administrative assistance.
Last but not least I would like to thank my parents for their love, constant support, long-
hours Skype calls and faith in me. You thought me that in life everything is possible. You
mean so much to me.
Without You all, I would never be where I am now.
Thank you.
ivAbstract
Since decades modern technological applications lead to challenging PDE eigenvalue prob-
lems, e.g., vibrations of structures, modeling of photonic gap materials, analysis of the hy-
drodynamic stability, or calculations of energy levels in quantum mechanics [6, 58, 78, 95].
Recently, a lot of research is devoted to the so-called Adaptive Finite Element Methods
(AFEM) [10]. In most AFEM approaches it is assumed that the resulting finite dimensional
algebraic problem (linear system or eigenvalue problem) is solved exactly and computational
costs for this part of the method as well as the fact that they are solved in finite precision
arithmetic are typically ignored.
The goal of this work is to analyze the influence of the accuracy of the algebraic approx-
imation on the adaptivity process. Efficient and reliable adaptive algorithms should take
into consideration not only discretization errors, but also iteration errors and especially for
non-symmetric problems the conditioning of the eigenvalues.
Our new AFEMLA algorithm extends the standard AFEM approaches to incorporate ap-
proximation errors into the adaptation process. Furthermore, we show that the adaptive
mesh refinement may be steered by the discrete residual vector, e.g., when the problem is
stated in a discrete formulation where only the underlying matrices and meshes are avail-
able. Moreover, we discuss how to reduce the computational effort of the iterative solver by
adapting the size ofthe Krylov subspace. With classical perturbation results we prove upper
bounds for the eigenvalue and the eigenfunction error. Under certain assumptions similar
results are obtained forconvection-diffusion problems. Following[9], we introduce functional
perturbation results for PDE eigenvalue problems including the functional backward error
and the functional condition number. These results are used to establish a combined a pos-
teriori error estimator embodying the discretization and the approximation error. Based on
perturbation results in the - and -norm derived in [65] and a standard resid-
ual a posteriori error estimator a balanced AFEM algorithm is proposed. The eigensolver
stopping criterion is based on the equilibrating strategy, i.e., iterations proceed as long as
the discrete part of the error estimator dominates the continuous part. A completely new
approach combining the adaptive finite element method with the homotopy method is in-