Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDEs on Tensor-Product Domains
Roland Pabel
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Roland Pabel, Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDEs on Tensor-Product Domains (2015), Logos Verlag, Berlin, ISBN: 9783832591595
Beschreibung / Abstract
This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the ell_2sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs.
Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.
Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.
Inhaltsverzeichnis
- BEGINN
- 1 Fundamentals
- 1.1 Basic Definitions and Vocabulary
- 1.2 Sobolev Spaces
- 1.3 Besov Spaces
- 1.4 Elliptic Partial Differential Equations
- 1.5 Nonlinear Elliptic Partial Differential Equations
- 2 Multiresolution Analysis and Wavelets
- 2.1 Multiscale Decompositions of Function Spaces
- 2.2 Multiresolutions of L2 and Hs
- 2.3 B-Spline Wavelets on the Interval
- 2.4 Multivariate Wavelets
- 2.5 Full Space Discretizations
- 3 Adaptive Wavelet Methods based upon Trees
- 3.1 Introduction
- 3.2 Nonlinear Wavelet Approximation
- 3.3 Algorithms for Tree Structured Index Sets
- 3.4 Application of Semilinear Elliptic Operators
- 3.5 Application of Linear Operators
- 3.6 Trace Operators
- 3.7 Anisotropic Adaptive Wavelet Methods
- 4 Numerics of Adaptive Wavelet Methods
- 4.1 Iterative Solvers
- 4.2 Richardson Iteration
- 4.3 Gradient Iteration
- 4.4 Newton†™s Method
- 4.5 Implementational Details
- 4.6 A 2D Example Problem
- 4.7 A 3D Example Problem
- 5 Boundary Value Problems as Saddle Point Problems
- 5.1 Saddle Point Problems
- 5.2 PDE Based Boundary Value Problems
- 5.3 Adaptive Solution Methods
- 5.4 A 2D Linear Boundary Value Example Problem
- 5.5 A 2D Linear PDE and a Boundary on a Circle
- 5.6 A 2D Nonlinear Boundary Value Example Problem
- 5.7 A 3D Nonlinear Boundary Value Example Problem
- 6 Résumé and Outlook
- 6.1 Conclusions
- 6.2 Future Work
- A Wavelet Details
- B Implementational Details
- C Notation