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    Optimal adaptive wavelet methods for solving first order system least squares

    Rekatsinas, N.

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    Citation for published version (APA): Rekatsinas, N. (2018). Optimal adaptive wavelet methods for solving first order system least squares.

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  • Optimal Adaptive Wavelet Methods

    for solving First Order SystemLeast Squares

    Nikolaos Rekatsinas

    O p t im

    a l A

    d a p t iv

    e W

    a v e le

    t M

    e t h o d s

    for solving

    F ir

    s t

    O r d e r

    S y s t e m

    L e a s t

    S q u a r e s

    N ik

    o la

    o s

    R e k a t s in

    a s

    Optimal Adaptive Wavelet Methods

    for solving First Order SystemLeast Squares

    Nikolaos Rekatsinas

    O p t im

    a l A

    d a p t iv

    e W

    a v e le

    t M

    e t h o d s

    for solving

    F ir

    s t

    O r d e r

    S y s t e m

    L e a s t

    S q u a r e s

    N ik

    o la

    o s

    R e k a t s in

    a s

  • Optimal Adaptive Wavelet Methods for solving First Order System Least Squares

    Nikolaos Rekatsinas

  • Optimal Adaptive Wavelet Methods for solving First Order System Least Squares

    ACADEMISCH PROEFSCHRIFT

    ter verkrijging van de graad van doctor

    aan de Universiteit van Amsterdam

    op gezag van de Rector Magnificus

    prof. dr. ir. K.I.J. Maex

    ten overstaan van een door het College voor Promoties

    ingestelde commissie,

    in het openbaar te verdedigen in de Agnietenkapel

    op donderdag 31 mei 2018, te 12:00 uur

    door

    Nikolaos Rekatsinas geboren te Marousi, Griekenland

  • Promotiecommissie

    Promotor: prof. dr. R.P. Stevenson (Universiteit van Amsterdam) Copromotor: dr. J.H. Brandts (Universiteit van Amsterdam)

    Overige leden: prof. dr. H. Harbrecht (University of Basel) prof. dr. A.J. Homburg (Universiteit van Amsterdam) prof. dr. T. Raasch (University of Siegen) dr. C.C. Stolk (Universiteit van Amsterdam) prof. dr. J.J.O.O. Wiegerinck (Universiteit van Amsterdam)

    Faculteit der Natuurwetenschappen, Wiskunde en Informatica

    This research was funded by NWO grant 613.001.216.

  • The illustration of the cover, and that of the dedication page is composed with parts of Lissitzky ’s book for children of all ages, Pro dva kvadrata (About two squares).

  • «Και πρώτα απ’ όλα τι εννοούμε λέγοντας παιδεία; Την πληροφορία, την τεχνική, το δίπλωμα εξειδίκευσης που εξ- ασφαλίζει γάμο, αυτοκίνητο κι ακίνητο, με πληρωμή την πλήρη υποταγή του εξασφαλισθέντος ή την πνευματική και ψυχική διάπλαση ενός ελεύθερου ανθρώπου, με τεχνική αναθεώρησης κι ονειρικής δομής, με αγωνία απελευθέρωσης και με διαθέσεις μιας ιπτάμενης φυγής προς τ’ άστρα;»

    Μάνος Χατζιδάκις

    "But first of all, what do we mean by education? The information, the technique, or the diploma of specialisa- tion, being potentially the guarantee of a marriage, mov- able and immovable property, that comes along with the price of full subordination of the person seeking this kind of security? Or, the spiritual and mental structure of a free man who practices the art of revision, and composed by the very essence of dreams craves after the agony of liberation for a flying escape to the stars?"

    Manos Hadjidakis

  • Contents

    1 Introduction 1

    1.1 The numerical solution of operator equations . . . . . . . . . . . . . . 1 1.1.1 Adaptive Wavelet Galerkin Methods . . . . . . . . . . . . . . . 2 1.1.2 Contributions from this thesis . . . . . . . . . . . . . . . . . . . 4

    2 An optimal adaptive wavelet method for first order system least

    squares 7

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Adaptive wavelet schemes, and the approximate residual evalu-

    ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 An alternative for the APPLY routine . . . . . . . . . . . . . 9 2.1.3 A common first order system least squares formulation . . . . . 11 2.1.4 A seemingly unpractical least squares formulation . . . . . . . 13 2.1.5 Layout of the chapter . . . . . . . . . . . . . . . . . . . . . . . 15

    2.2 Reformulation of a semi-linear second order PDE as a first order system least squares problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.3 The adaptive wavelet Galerkin method (awgm) . . . . . . . . . . . . . 20 2.4 Application to normal equations . . . . . . . . . . . . . . . . . . . . . 23 2.5 Semi-linear 2nd order elliptic PDE . . . . . . . . . . . . . . . . . . . . 24

    2.5.1 Reformulation as a first order system least squares problem . . 24 2.5.2 Wavelet assumptions and definitions . . . . . . . . . . . . . . . 26 2.5.3 An appropriate approximate residual evaluation . . . . . . . . 31

    2.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Stationary Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . 39

    2.7.1 Velocity–pressure–velocity gradient formulation . . . . . . . . . 41 2.7.2 Velocity–pressure–vorticity formulation . . . . . . . . . . . . . 43

    2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.9 Appendix: Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . 46

    3 A quadratic finite element wavelet Riesz basis 51

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Theory on biorthogonal wavelet bases . . . . . . . . . . . . . . . . . . 53

  • 0. CONTENTS

    3.3 Construction of quadratic Lagrange finite element wavelets . . . . . . 57 3.3.1 Multi-resolution analyses . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 Local-to-global basis construction . . . . . . . . . . . . . . . . . 58 3.3.3 Verification of the uniform inf-sup conditions for (Vj , Ṽj)j≥0 . . 60 3.3.4 Local collections Θ, Ξ, and Φ̃ underlying the construction of Ψj+1 60 3.3.5 Definition of the Ψj . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.6 Condition numbers . . . . . . . . . . . . . . . . . . . . . . . . . 63

    4 An optimal adaptive tensor product wavelet solver of a space-time

    FOSLS formulation of parabolic evolution problems 67

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Well-posed FOSLS formulation of a parabolic PDE . . . . . . . . . . . 70 4.3 The adaptive wavelet Galerkin method (awgm) . . . . . . . . . . . . . 73 4.4 Application to the FOSLS formulation . . . . . . . . . . . . . . . . . . 76

    4.4.1 Expression for the residual . . . . . . . . . . . . . . . . . . . . 76 4.4.2 Tensor product bases . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.3 Piecewise polynomial spatial and temporal wavelets . . . . . . 78 4.4.4 Alpert wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4.5 Multi-tree approximation . . . . . . . . . . . . . . . . . . . . . 81 4.4.6 Best possible rate . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4.7 Constructing the approximate residual . . . . . . . . . . . . . . 83

    4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7 Appendix: Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . 92

    5 On the awgm implementation 101

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Wavelets bases: A brief presentation of collections in use . . . . . . . . 103

    5.2.1 Haar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2.2 Orthonormal discontinuous piecewise linear wavelets . . . . . . 103 5.2.3 Continuous piecewise linear 3-point wavelets in 1D . . . . . . . 104 5.2.4 Continuous piecewise linear 3-point wavelets in 2D . . . . . . . 105 5.2.5 Continuous piecewise quadratic wavelets in 2D . . . . . . . . .