The Semismooth Newton Method for the Solution of Reactive Transport Problems Including Mineral Precipitation-Dissolution Reactions [Elektronische Ressource] / Hannes Buchholzer. Betreuer: Christian Kanzow

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• Mathematics
• Physik

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• Publié par: julius-maximilians-universitat_wurzburg
• Publié le: 01 janvier 2011
• Nombre de lectures: 4
• Langue: English

Extrait

The Semismooth Newton Method
for the Solution of Reactive
Transport Problems Including
Mineral Precipitation-Dissolution
Reactions
Hannes Buchholzer
Dissertation
University of Würzburg
Department of MathematicsThe Semismooth Newton Method
for the Solution of Reactive
Transport Problems Including
Mineral Precipitation-Dissolution
Reactions
Dissertation zur Erlangung
des naturwissenschaftlichen Doktorgrades
der Julius-Maximilians-Universität Würzburg
vorgelegt von
Hannes Buchholzer
aus
Hermannstadt
Eingereicht am: 2. Februar 2011
1. Gutachter: Prof. Dr. Christian Kanzow, Universität Würzburg
2. Dr. habil. Serge Kräutle, Universität Erlangen-NürnbergAcknowledgment
This present doctoral thesis is the result of over 3 years of research that I conducted at
the Department of Mathematics at the University of Würzburg. I would like to seize this
opportunity to express my thanks to some people who supported me during this time either
scientifically or personally.
First of all there is my supervisor Christian Kanzow. He almost always took time (inter-
rupting his present work) to answer questions. Furthermore he provided guidance, dis-
cussed ideas, had valuable suggestions and co-authored two articles with me. I could
greatly benefit from his experience and knowledge. Also he was (and is) a pleasant person
with whom I enjoyed working. Finally he even supported me in my struggle against En-
glish grammar and this and other publications. For all of this I want to express my deep
gratitude to him.
This research project was a cooperation between the chair of Applied Mathematics II
at the University of Würzburg under the head of C. Kanzow and the chair of Applied
Mathematics I at the University of Erlangen-Nuremberg under the head of P. Knabner.
While the work group in Erlangen provided the application from the field of hydrogeology
together with all their knowledge and experience in this area, it was the task of the work
group in Würzburg to contribute their e in the field of optimization to this matter.
In particular it was my task to investigate the mentioned application from the viewpoint of
modern optimization. My main contact person from the work group in Erlangen was Serge
Kräutle. He took the time to answer my questions with very comprehensive Emails that I
could understand better than corresponding technical literature. From him I really learned
very much about the main application and topics from neighboring fields like physics,
chemistry and geoscience. I want to express my sincere gratitude for this and also for
being a friendly pleasant person.
Furthermore I would like to thank my father for supporting me financially during my
studies, which in the first place enabled me to come this far. Also I thank him for his
readiness to help me otherwise as far as he could.
Last but not least I would like to thank my best friend for his support and friendship.
This is invaluable to me.viContents
1 Introduction 1
2 Preliminaries 5
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Subdierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Semismooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Complementary Problems and NCP-Functions . . . . . . . . . . . . . . . . 15
2.5 Newton Method for Semismooth Functions . . . . . . . . . . . . . . . . . 18
3 ProblemFormulationandTransformation 21
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Overall Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Index of Technical Terms . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.3 Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.4 Chemical . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Transformation of the Dynamic System . . . . . . . . . . . . . . . . . . . 32
3.4 Discretization of the Dynamic System . . . . . . . . . . . . . . . . . . . . 33
4 TheMinimumFunctionApproach 35
4.1 Study of Subdierentials of G . . . . . . . . . . . . . . . . . . . . . . . 35M
4.2 Newton’s Method and Active Set Strategy . . . . . . . . . . . . . . . . . . 39
4.3 More Subdierentials and their Transformations . . . . . . . . . . . . . . . 41
4.4 First Proof of Nonsingularity . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 New Proof of . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Schur Complement Approach . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 Existence and Uniqueness of a Local Solution . . . . . . . . . . . . . . . . 59
4.8 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 TheFischer-BurmeisterFunctionApproach 69
5.1 Study of Subdierentials of G . . . . . . . . . . . . . . . . . . . . . . . . 70F
5.2 Local Convergence and Nonsingularity . . . . . . . . . . . . . . . . . . . . 72
5.3 Globalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 TheDeterminantTheory 83
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Contents viii
6.2 Formulas for the Entries of F(y) . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 Bounds for F(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Application to the Main Problem . . . . . . . . . . . . . . . . . . . . . . . 94
7 TheSingularValueTheory 99
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Estimates for the Extremal Eigenvalues . . . . . . . . . . . . . . . . . . . 104
7.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 FinalRemarks 123
A Appendix 125
A.1 Determinant Sum Expansion Formula . . . . . . . . . . . . . . . . . . . . 125
A.2 The and Block Permutations . . . . . . . . . . . . . . . . . . 127
A.3 Results for the Spectral Norm . . . . . . . . . . . . . . . . . . . . . . . . 129
RA.4 Calculating the Bound for F(y) with MATLAB . . . . . . . . . . . . . . 1301 Introduction
In the fundamental and vital papers [39] by Qi and Sun, and [38] by Qi semismooth New-
ton methods where introduced for the solution of nonlinear systems of equations that are
defined by a nonsmooth mapping. Related material can also be found in Kummer [29, 30].
Similar to the classical Newton method these methods can be shown to be locally quadrati-
cally convergent under moderate assumptions that are somewhat similar to the assumptions
that the classical method requires. Since the introductory publications from Qi and Sun,
semismooth Newton methods have been used to solve many classes of mathematical prob-
lems that can be formulated in a suitable way as a nonsmooth system of equations, among
them are complementary problems, variational inequalities and nonsmooth equation sys-
tems themselves, cf. [12a, 12b, 34]. In these days they are widely accepted, used and
studied.
In this thesis we consider a particular class of applications from the field of compu-
tational geosciences, namely a reactive transport model in the subsurface including min-
eral precipitation-dissolution reactions. This model involves partial dierential equations
(PDEs), ordinary dierential equations (ODEs), and algebraic equations, together with
some complementarity conditions arising from the equilibrium conditions of the minerals.
After discretization this results in a mixed complementary problem that can be equivalently
written as a nonlinear and nonsmooth system of equations via so-called NCP-functions.
This seems to be a new and promising approach in literature for problems of this kind.
In general, the modeling of reactive transport problems in porous media leads to systems
involving PDEs and ODEs; the PDEs for the concentration of chemical species which are
dissolved in the water (called mobile species), and the ODEs for the concentrations of
species which are attached to the soil matrix and which are not subject to transport (called
immobile species). In the following we assume that all of the immobile species, from a
chemist’s point of view, are minerals. Hence, the reactions with minerals are so-called
precipitation-dissolution reactions.
In principle, it is possible to model reactions among the mobile and between mobile and
immobile species by kinetic rate laws, i.e., the reactive source/sink terms in the PDEs and
ODEs are given functions of the concentrations, coupling the PDEs and ODEs. If the re-
actions are suciently fast, then the assumption of local equilibrium instead is reasonable.
This equilibrium assumption means that the concentrations of the involved species tend to a
certain reaction dependent ratio between reactants and products. For reactions among mo-
bile species, local equilibrium conditions can be described by (nonlinear) algebraic equa-
tions (AEs). However for reactions involving minerals, complementary conditions (CCs)
are necessary to describe local equilibrium. Because in this case another state of equilib-
rium is possible: complete dissolution of