Note: Literaturverzeichnis: Seite 12

Note: Intro -- Preface -- Contents -- Part I Scaling Limits of Diffusion Processes and Singular Spaces -- Ricci Bounds for Euclidean and Spherical Cones -- 1 Introduction -- 1.1 Statement of the Main Results -- 1.2 Basic Definitions and Notations -- 1.3 The Curvature-Dimension Condition -- 2 Euclidean Cones over Metric Measure Spaces -- 3 Spherical Cones over Metric Measure Spaces -- References -- A Monotone Approximation to the Wasserstein Diffusion -- 1 Introduction and Statement of the Main Results -- 2 Dirichlet-Ferguson Process, Entropic Measure and Wasserstein Diffusion -- 2.1 The Dirichlet-Ferguson Process -- 2.2 The Dirichlet Form on G -- 2.3 The Dirichlet Form on the Wasserstein Space -- 3 The Distribution of Random Means -- 4 The Measure mkβ in the Multivariate Case -- 5 Projections, Isomorphisms, Approximations -- 5.1 Finite Dimensional Projections -- 5.2 Monotone Convergence -- 5.3 Isomorphisms I -- 5.4 Standard Approximations -- 5.5 Isomorphisms II -- 5.6 Isomorphisms III -- 6 Convergence -- 6.1 Convergence of Finite Dimensional Distributions -- 6.2 Convergence of Processes -- 6.3 Final Remarks -- References -- Adapted Function Spaces for Dispersive Equations -- 1 Introduction -- 2 The Bounded p Variation -- 2.1 Step Functions and Ruled Functions -- 2.2 The Spaces Vp and Up -- 2.3 Embeddings -- 3 Dispersive Equations -- 3.1 Adapted Function Spaces -- 3.2 The Generalized KdV Equation -- 3.3 The Kadomtsev-Petviashvili II Equation -- 3.4 The Energy Critical Nonlinear Schrrödinger Equation on Compact Manifolds -- References -- A Note on Metastable Behaviour in the Zero-Range Process -- 1 Introduction -- 1.1 The Model and Basic Properties -- 2 Metastability -- 2.1 Definitions and Results -- 3 Results for the Zero-Range Process -- 4 Proofs of the Results -- 4.1 Capacity -- 4.1.1 Lower Bound -- 4.1.2 Upper Bound -- 4.2 Proofs of Theorems 3 and 4. , 4.3 Metastability of the Zero-Range Process -- References -- Convergence of the Two-Point Function of the Stationary TASEP -- 1 Introduction and Result -- 2 Setting and Strategy of the Proof -- 3 Proof of Proposition 1: Upper Tail -- 3.1 Asymptotics for -- 3.2 Evaluation of g1 -- 3.3 Estimations on g2 and g3 -- 4 Proof of Proposition 2: Lower Tail -- 5 Proof of Corollary 2 -- References -- Part II Multiple Scales in Mathematical Models of Materials Science and Biology -- Vortex Motion for the Landau-Lifshitz-Gilbert Equation with Applied Magnetic Field -- 1 Introduction -- 2 Jacobian, Vorticity and Renormalized Energy -- 2.1 Compactness -- 2.2 The Renormalized Energy -- 2.3 The Thiele Equation -- 3 LLG Equation with External Fields -- 3.1 Conservation Laws -- 3.2 Weak Solutions and Bubbling -- 4 Convergence and Vortex Trajectories -- 5 Motion Law -- References -- On Prandtl-Reuss Mixtures -- 1 Introduction -- 2 Mathematical Formulation of the Problem -- 2.1 Basic Setting -- 2.2 Balance of Linear Momentum -- 2.3 Elasticity -- 2.4 The Variational Inequality for Elastic-Perfect-Plastic Mixtures -- 3 The Penalty Equation -- 3.1 Reconstruction of Partial Strains -- 4 The Rothe Approximation -- 4.1 Definition and Solvability of the Rothe Approximation -- 4.2 First Estimates for the Rothe Approximation -- 5 Estimates for the Rothe Approximation -- 6 Convergence of the Rothe Method -- 7 L∞(L2)-Estimate for the Time Derivatives of the Stresses -- 8 Passage to the Limit as the Penalty Parameter μ Tends to Zero -- 9 A Model for the Volume Fraction α and the Yield Parameter Depending on the History of the Rate of the Plastic Strain of the Soft Material -- References -- Modeling and Simulation of Lipid Monolayers as Surfactant in Lung Alveoli -- 1 Biological Function of Surfactant Lipids for the Breathing Cycle -- 2 Microscale Stochastic Multi-particle Model. , 3 Mesoscale Continuum Mixture Model -- 4 Sharp Phase Transition Approximation -- 4.1 Case of Dissolving Ordered Monolayer: λ> -- 0 -- 4.2 Case of Growing Ordered Monolayer: λ< -- 0 -- 4.3 Simulation Results -- 5 One-Dimensional Simulation of the Continuum Phase Field Model -- 5.1 Smoothed Phase Separation in a Fixed Interval -- 5.2 Phase Separation Behavior in an Oscillating Domain Simulating ``Breathing'' -- 6 Conclusions -- References -- Fluid Mixtures and Applications to Biological Systems -- 1 Introduction -- 2 Fluid Mixtures -- 3 Exploiting the Free Energy Inequality -- 4 Remark on Pressure -- 5 Examples -- 6 Handling Gradient Terms -- 7 Polymerization of Actin Filaments -- 8 The Quasistatic Problem -- 9 Fractional Densities -- 10 Diffusion Limit -- 11 Polymer Mixtures Including Gradients -- 12 Conclusion -- References -- A Nested Variational Time Discretization for Parametric Anisotropic Willmore Flow -- 1 Introduction -- 2 Review of the Time Discretization of Isotropic Willmore Flow -- 3 Nested Time Discretization for Anisotropic Willmore Flow -- 4 Finite Element Discretization in Space -- 5 Optimization Algorithm for the Time Steps -- 6 Numerical Results -- References -- Energy Scaling and Domain Branching in Solid-Solid PhaseTransitions -- 1 Introduction -- 2 Upper Bounds -- 2.1 The Case of Two Rank-One Connections -- 2.2 The Case of One Rank-One Connection -- 3 Lower Bound -- 3.1 The Case of Two Rank-One Connections -- 3.2 The Case of One Rank-One Connection -- 4 Outlook -- References -- Part III Numerics for Multiscale Models and Singular Phenomena -- On a Multilevel Preconditioner and its Condition Numbersfor the Discretized Laplacian on Full and Sparse Grids in HigherDimensions -- 1 Introduction -- 2 Discretization -- 2.1 Discretization by an Isotropic Full-Grid -- 2.2 The Multilevel Approach. , 2.3 A Norm Equivalence Based on Orthogonal Subspaces -- 3 A Dimension-Independent Full Grid Preconditioner -- 3.1 Orthogonalization Operator -- 3.2 Preconditioner -- 3.3 Cost Discussion for the New Preconditioner -- 4 Sparse Grids -- 4.1 Definition -- 4.2 Sparse Grid Submatrix and Preconditioner -- 4.3 Cost Discussion -- 5 Prewavelets -- 5.1 Preconditioner -- 5.2 Cost Discussion -- 6 Numerical Experiments -- 7 Concluding Remarks -- References -- Simulation of Droplet Impact with Dynamic Contact Angle Boundary Conditions -- 1 Introduction -- 2 The Moving Contact Line Problem -- 2.1 Modeling Issues -- 2.2 Numerical Issues -- 3 Mathematical Model -- 3.1 The Navier-Stokes Solver -- 3.2 The Contact Angle as a Boundary Condition -- 3.3 The Dynamic Contact Angle Model by Yokoi et al. (C1) -- 3.4 The Dynamic Contact Angle Model by Shikhmurzaev (C2) -- 4 The Numerical Method -- 4.1 Discretization of the Navier-Stokes Equations and the Level-Set Method -- 4.2 Discretization of the Contact Angle Boundary Condition -- 4.3 Implementation of the Contact Angle Models -- 4.4 Methods for Mass Conservation -- 5 Numerical Results -- 5.1 The Contact Angle Model by Yokoi -- 5.2 The Contact Angle Model by Shikhmurzaev -- 5.3 Mass Conservation -- 6 Conclusion -- References -- A Parallel Multiscale Simulation Toolbox for Coupling Molecular Dynamics and Finite Elements -- 1 Introduction -- 1.1 Related Work -- 1.2 Article Contribution and Outline -- 2 Multiscale Simulation Method -- 2.1 Molecular Dynamics -- 2.2 Continuum Mechanics and Finite Elements -- 2.3 Coupling Method -- 2.3.1 Coupling with Overlap -- 2.3.2 Damping High Fluctuation Modes -- 2.3.3 Complete Algorithm -- 3 Multiscale Simulation Toolbox -- 3.1 Interface Design -- 3.1.1 Opaque Handles -- 3.1.2 Access Epochs -- 3.1.3 Piggybacking of Metadata -- 3.2 Description of the Maci Code -- 4 Parallelization Aspects. , 4.1 Challenges -- 4.2 Data Distribution in the Maci Code -- 4.3 Parallel Assembly -- 4.4 Runtime Support -- References -- A Moving Least Squares Approach to the Construction of Discontinuous Enrichment Functions -- 1 Introduction -- 2 Particle Data -- 3 The Moving Least Squares Method -- 4 Adjacency Based Modification -- 5 Proof of Concept -- 6 Concluding Remarks -- References -- Second Moment Analysis for Robin Boundary Value Problems on Random Domains -- 1 Introduction -- 2 Robin Boundary Value Problems on Random Domains -- 3 Variational Formulation -- 4 Solving Tensor Product Boundary Value Problems -- 4.1 An Abstract View on the Linearization Approach -- 4.2 Sparse Tensor Product Spaces -- 4.3 Sparse Multilevel Frames -- 4.4 Combination Technique -- 4.5 Low-Rank Approximation -- 5 Finite Element Discretization -- 5.1 Parametric Finite Elements -- 5.2 Galerkin Discretization -- 5.3 Implementation of the Combination Technique -- 5.4 Implementation of the Low-Rank Approximation -- 6 Numerical Results -- 6.1 Model Verification -- 6.2 A Correlation Kernel of Arbitrary Smoothness -- 6.3 A Correlation Kernel of Finite Smoothness -- 7 Concluding Remarks -- References -- Simulation of Q-Tensor Fields with Constant Orientational Order Parameter in the Theory of Uniaxial Nematic Liquid Crystals -- 1 Introduction and Derivation of the Mathematical Setting -- 2 Euler Lagrange Equations for ELdG and EOF -- 3 Discrete Setting -- 3.1 Monotonicity Estimates -- 3.2 Euler Lagrange Equation in the Discrete Setting -- 4 Iterative Algorithms -- 4.1 Fully Discrete Algorithm for Discrete Harmonic Director Fields -- 4.2 Fully Discrete Algorithm for Discrete Q Harmonic Tensor Fields -- 5 Analysis of the Algorithms and Q Harmonic Tensor Fields -- 5.1 Stability and Convergence of the Tensor Field Algorithm -- 5.2 Weak Compactness Result for Q Harmonic Tensor Fields. , 6 Numerical Experiments.