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Table of Contents
Intro
Preface
Contents
Editors and Contributors
About the Editors
Contributors
A Numerical Approach to a Nonlinear Diffusion Model for Self-Organized Criticality Phenomena
1 Introduction
2 The Basic Model
3 Numerical Approximation on a Fixed Grid
4 Approximation on a Synchronized Family of Grids
5 Numerical Tests
References
Approximation of 3D Stokes Flows in Fractal Domains
1 Introduction
2 Preliminaries
3 Existence and Uniqueness Results
4 Regularity in Weighted Sobolev Spaces
5 Mean Shear Stress
6 Numerical Approximation
7 Numerical Simulations
References
∞-Laplacian Obstacle Problems in Fractal Domains
1 Introduction
2 Fractal Domains, Approximating Domains and Fibers
3 Setting and Asymptotic Behavior
4 Uniqueness and Perspectives
5 Uniform and Error Estimates
References
Discretization of the Koch Snowflake Domain with Boundary and Interior Energies
1 Introduction
2 Dirichlet Form on the Koch Snowflake
3 Dirichlet Form on the Snowflake Domain
4 Inductive Mesh Construction and Discrete Energy Forms
5 Numerical Results
5.1 Algorithm and Implementation
5.2 The Eigenvalue Counting Function
5.3 Eigenvectors and Eigenvalues in the Low Eigenvalue Regime
5.4 Localization in the High Eigenvalue Regime
6 A Landscape Approach to High Frequency Localization
References
On the Dimension of the Sierpinski Gasket in l2
1 Introduction
2 Invariant Sets
2.1 Infinite Dimensional Sierpinski Gasket
2.2 Hausdorff Dimension of Invariant Sets
2.3 N-Dimensional Simplices
3 Invariant Measures
References
On the External Approximation of Sobolev Spaces by M-Convergence
1 Introduction
2 Sobolev Space Approximations
3 The M-Convergence Result
4 Proof of Lemma 1
5 Proof of Lemma 2
6 Comments
References
Generalization of Rellich-Kondrachov Theorem and Trace Compactness for Fractal Boundaries
1 Introduction
2 Sobolev Extension Domains
3 Trace on the Boundary and Green Formulas
3.1 Framework of d-Sets and Markov's Local Inequality
3.2 General Framework of Closed Subsets of Rn
3.3 Integration by Parts and the Green Formula
4 Sobolev Admissible Domains and the Generalization of the Rellich-Kondrachov Theorem
5 Compactness of the Trace
6 Application to the Poisson Boundary Valued and Spectral Problems
References
Preface
Contents
Editors and Contributors
About the Editors
Contributors
A Numerical Approach to a Nonlinear Diffusion Model for Self-Organized Criticality Phenomena
1 Introduction
2 The Basic Model
3 Numerical Approximation on a Fixed Grid
4 Approximation on a Synchronized Family of Grids
5 Numerical Tests
References
Approximation of 3D Stokes Flows in Fractal Domains
1 Introduction
2 Preliminaries
3 Existence and Uniqueness Results
4 Regularity in Weighted Sobolev Spaces
5 Mean Shear Stress
6 Numerical Approximation
7 Numerical Simulations
References
∞-Laplacian Obstacle Problems in Fractal Domains
1 Introduction
2 Fractal Domains, Approximating Domains and Fibers
3 Setting and Asymptotic Behavior
4 Uniqueness and Perspectives
5 Uniform and Error Estimates
References
Discretization of the Koch Snowflake Domain with Boundary and Interior Energies
1 Introduction
2 Dirichlet Form on the Koch Snowflake
3 Dirichlet Form on the Snowflake Domain
4 Inductive Mesh Construction and Discrete Energy Forms
5 Numerical Results
5.1 Algorithm and Implementation
5.2 The Eigenvalue Counting Function
5.3 Eigenvectors and Eigenvalues in the Low Eigenvalue Regime
5.4 Localization in the High Eigenvalue Regime
6 A Landscape Approach to High Frequency Localization
References
On the Dimension of the Sierpinski Gasket in l2
1 Introduction
2 Invariant Sets
2.1 Infinite Dimensional Sierpinski Gasket
2.2 Hausdorff Dimension of Invariant Sets
2.3 N-Dimensional Simplices
3 Invariant Measures
References
On the External Approximation of Sobolev Spaces by M-Convergence
1 Introduction
2 Sobolev Space Approximations
3 The M-Convergence Result
4 Proof of Lemma 1
5 Proof of Lemma 2
6 Comments
References
Generalization of Rellich-Kondrachov Theorem and Trace Compactness for Fractal Boundaries
1 Introduction
2 Sobolev Extension Domains
3 Trace on the Boundary and Green Formulas
3.1 Framework of d-Sets and Markov's Local Inequality
3.2 General Framework of Closed Subsets of Rn
3.3 Integration by Parts and the Green Formula
4 Sobolev Admissible Domains and the Generalization of the Rellich-Kondrachov Theorem
5 Compactness of the Trace
6 Application to the Poisson Boundary Valued and Spectral Problems
References