Geometric Partial Differential Equations - Part I

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Contributors
Preface
	Acknowledgements
Finite element methods for the Laplace-Beltrami operator
	Introduction
	Calculus on surfaces
		Parametric surfaces
		Differential operators
		Signed distance function
		Curvatures
		Surface regularity and properties of the distance function
		Divergence theorem on surfaces
	Perturbation theory
		Perturbation theory for C1,α surfaces
		Perturbation theory for C2 surfaces
		H2 extensions from C2 surfaces
			Regularization
	Parametric finite element method
		FEM on Lipschitz parametric surfaces
			Lipschitz parametric surfaces
			Differential geometry on polyhedral surfaces
			Parametric finite element method
		Geometric consistency
			Uniform Poincaré-Friedrichs estimate on Γ
			Geometric estimators
			Geometric consistency error for C1,α surfaces
			Geometric consistency error for C2 surfaces
		A priori error analysis
			A priori error estimates for C2 surfaces
			A priori error estimates for C1,α surfaces
		A posteriori error analysis
	Trace method
		Preliminaries
			Bulk and surface meshes
			Geometric assumptions
			Level set representations
			Harmonic extension and traces
		A priori error estimates
			Geometric resolution and extensions
			Approximation properties of trace finite element space
		A posteriori error estimates
			Assumptions on surface representation
			Notation and surface resolution assumptions
			Extension for a posteriori error estimates
			Preliminary results
			A posteriori upper bound
	Narrow band method
		The narrow band FEM
		PDE geometric consistency
		Properties of the narrow band FEM
		A priori error estimates
	Acknowledgements
	References
The Monge-Ampère equation
	Introduction
		Geometric applications
			Gauss curvature problem
			Reflector design problem
			Affine plateau problem
			Optimal mass transport problem
		Solution concepts for the Monge-Ampère equation
			Classical solutions
			Viscosity solutions
			Alexandrov solutions
	Wide stencil finite differences
		A general framework for approximation schemes
		A variational characterization of the determinant
		Wide stencil finite difference schemes
		Filtered schemes
		Lattice basis reduction scheme
		Discretization based on power diagrams
		Two scale methods
			Discrete convexity
			A comparison principle
			Consistency and discrete barriers
			Convergence
			Rates of convergence
		Extensions, generalizations, and applications
			Hamilton Jacobi Bellman formulation and semi-Lagrangian schemes
			Filtered two scale schemes
			Approximation of convex envelopes
			The Gauss curvature problem
			Transport boundary conditions
	Discretizations based on geometric considerations
		Description of the scheme
			Nodal set and domain partition
			Nodal functions, their subdifferentials, and convex envelopes
			The Oliker-Prussner method
		Stability, continuous dependence on data, and discrete maximum principle
		Consistency
		Pointwise error estimate
		W2,p error estimate
	Finite Element Methods
		Continuous finite element methods
		Mixed formulations
		Galerkin methods for singular solutions
			Convergence of interior discretizations
	Numerical examples
		Example 1: Smooth solution
		Example 2: Nonclassical solution
		Example 3: Lipschitz and degenerate solution
	Concluding remarks
	Acknowledgements
	References
Finite element simulation of nonlinear bending models for thin elastic rods and plates
	Introduction
		Bending of elastic rods
		Elastic plates
		Outline of the article
		Notation
	Formal dimension reductions
		Elastic rods
		Elastic plates
	Convergent finite element discretizations
		Elastic rods
		Elastic plates
	Iterative solution via constrained gradient flows
		Elastic rods
		Elastic plates
	Linear finite element systems with nodal constraints
		Application to harmonic maps
	Applications, modifications, and extensions
		Bilayer plates
		Self-avoiding curves and elastic knots
		Föppl-von Kármán model
	Conclusions
	Acknowledgements
	References
Parametric finite element approximations of curvature-driven interface evolutions
	Introduction
	Geometry of surfaces
		Surfaces in Rd
		Curvature
		The divergence theorem
		Evolving surfaces and transport theorems
		Time derivatives of the normal
		Time derivatives of the mean curvature
		Gauss-Bonnet theorem
	Parametric finite elements
		Polyhedral surfaces
			Orientation
			Polygonal curves
		Stability estimates
		Curvature approximations
		Evolving polyhedral surfaces and transport theorems
		Further results for evolving polyhedral surfaces
	Mean curvature flow
		Weak formulation
		Finite element approximation
		Discrete linear systems
			Curves in the plane
		Existence and uniqueness
		Stability
		Equipartition property
		Alternative parametric methods
			The classical Dziuk approach
			The convergent finite element algorithm of Kovács, Li, Lubich
			Alternative numerical methods that equidistribute
			Using the DeTurck trick to obtain good mesh properties
			Other numerical approaches
	Surface diffusion and other flows
		Properties of the surface diffusion flow
		Finite element approximation for surface diffusion
		Volume conservation for the semidiscrete scheme
		Generalizations to other flows
		Approximations with reduced or induced tangential motion
		Alternative parametric methods
	Anisotropic flows
		Derivation of the governing equations
		Suitable weak formulations
		Finite element approximation
		Solution method and discrete systems
		Volume conservation for semidiscrete schemes
		Alternative numerical approaches
	Coupling bulk equations to geometric equations on the surface and applications to crystal growth
		The Mullins-Sekerka problem
			Weak formulation of the Mullins-Sekerka problem
			An unfitted finite element approximation of the Mullins-Sekerka problem
		The Stefan problem with a (kinetic) Gibbs-Thomson law
		One-sided free boundary problems
		Alternative numerical approaches
	Two-phase flow
		Two-phase Stokes flow
			Finite element approximation
			Semidiscrete finite element approximation
			XFEMΓ for conservation of the phase volumes
			Approximations based on the fluidic tangential velocity
		Two-phase Navier-Stokes flow
		Alternative numerical approaches
	Willmore flow
		Derivation of the flow
		A finite element approximation of Willmore flow
		A stable approximation of Willmore flow
		Willmore flow with spontaneous curvature and area difference elasticity effects
		Alternative numerical approaches
	Biomembranes
		A model for the dynamics of fluidic biomembranes
		A weak formulation for the dynamics of biomembranes
		Semidiscrete finite element approximation
		Two-phase biomembranes
		Alternative numerical approaches
	Acknowledgement
	References
The phase field method for geometric moving interfaces and their numerical approximations
	Introduction
	Mathematical foundation of the phase field method
		Geometric surface evolution
		Examples of geometric surface evolution
		Mathematical formulations and methodologies
		Level set and phase field formulations of the MCF
		Phase field formulations of other moving interface problems
		Relationships between phase field and other formulations
		Phase function representations of geometric quantities
		Convergence of the phase field formulation
	Time-stepping schemes for phase field models
		Classical schemes
		Convex splitting and stabilized schemes
		Schemes using Lagrangian multipliers
		Further considerations
	Spatial discretization methods for phase field models
		Spatial finite difference discretization
		Spatial Galerkin discretizations
			Galerkin discretization via finite element methods
			Galerkin discretization via spectral methods
			Galerkin discretization via DG methods
			Isogeometric analysis
		Spatial mixed discretization
		Implementations and advantages of high order methods
	Convergence theories of fully discrete numerical methods
		Construction of fully discrete numerical schemes
		Types of convergence and a priori error estimates
		Coarse error estimates for a fixed value ε > 0
		Fine error estimates and convergence of numerical interfaces as ε, h, τ  0
	A posteriori error estimates and adaptive methods
		Spatial and temporal adaptivity
		Coarse and fine a posteriori error estimates for phase field models
	Applications and extensions
		Materials science applications
		Fluid and solid mechanics applications
		Image and data processing applications
		Biology applications
		Other variants of phase field models
			Nonlocal and factional order phase field models
			Stochastic phase field models
	Conclusion
	Acknowledgements
	References
	Further readings
A review of level set methods to model interfaces moving under complex physics: Recent challenges and advances
	Introduction
		Overview
		Outline of review article: The evolution of level set methods
	Level set methods: Background and formulation
		Formulation and equations of motion
		Advantages of this formulation
		Numerical approximations
	A first example: Geometry
	Narrow banding and extension velocities
		Narrow band level set methods
			Implementation of narrow band methods
			Reinitialization
			Modern reinitialization techniques
		Extension velocities
			Need for extension velocities
			Constructing extension velocities
		Transport and diffusion of material quantities on an evolving interface
	Applications of narrow band and extension methodologies
		Application of extension velocities: Two-phase flow and industrial inkjet printing
			Problem statement
			Equations of motion
			Algorithms and numerical implementation
				Meshing
				Wall contact angles, boundary conditions
				Dealing with viscoelasticity
				A comment about mass loss
				Computing implementation
			Physical parameters and results
		Application of extension physics: Micro-bubble dynamics
			Problem statement
			Equations of motion
			Algorithms and approximations
			Results
	Multi-phase physics: Mathematical formulation and algorithms for tracking multiple regions
		Beyond two phases
		Previous algorithms to handle multi-phase evolution
		The Voronoi Implicit Interface Method
		Motivation and fundamental idea of VIIM
		Algorithm flow
		Mathematical formulation
		Implementation of Voronoi reconstruction steps
	Applications of the Voronoi Implicit Interface Method
		Sintering and grain growth
		Variable density fluid flow
		Soap bubbles and industrial foams
			The dynamics of foam evolution
			A scale-separation model
			A mathematical model for foam evolution
	Sharp interface physics: Implicit mesh discontinuous Galerkin methods
		The challenge of sharp interface physics
			Smoothed interface methods
			Sharp interface methods
			Hybrid interface methods
		A high-order discontinuous Galerkin implicit mesh level set method for sharp interface physics
	Application: High-order DG implicit mesh level set methods for sharp interface fluid dynamics
	Summary
	Acknowledgement
	References
Free boundary problems in fluids and materials
	Introduction to geometric free boundary problems in fluids and materials
	Aspects of modelling
		Mathematical model of two-phase flow
			Setting and generic balance equations
			Incompressibility
			Conservation of momentum
			Jump condition at the interface
		Problem formulation of two-phase flow
		The Stefan problem with surface tension and kinetic undercooling
		The Stefan problem with capillary melt surface
		Phase variables and phase field models
			Enthalpy formulation of the Stefan problem
			Allen-Cahn equation and phase field models
	Numerical methods for two-phase flow
		Introduction
		Numerics for multiphase flow: Some general considerations and problems
		Mesh moving
			ALE formulation of the model problem
			Variational treatment of interface forces
			Mesh deformation
			Moving mesh method for the two-phase problem: Overall method
		Level set method for two-phase flow
			Pressure jump and XFEM
	Numerical methods for models with a parabolic interface equation
		Explicit treatment, parametric representation
		Implicit treatment, level set
			Anisotropic motion of level sets
		Implicit treatment, phase field
	Model problems, examples, and applications
		Uniaxial extensional flow in liquid bridges
			Problem setting
			Comparison of experimental and numerical results
			Parameter variations and bridge shapes
			Strain and shear
		Two-phase flow under microgravity without mass transfer
			Surface deformation by thermocapillary convection
			Reorientation behaviour of cryogenic liquids
		Material accumulation by melting and solidification
		Welded joints
		Dendritic solidification with and without flow in the liquid
			Sharp interface approach for dendritic growth
			Dendritic growth with convection in the melt
			Phase field approach
	Acknowledgements
	References
Discrete Riemannian calculus on shell space
	Introduction
	Related work
		Shape spaces
		Interpolation and extrapolation
		Riemannian splines
		Discrete variational methods
	Riemannian geometry and Hessian structure of shell space
		Brief recap of smooth Riemannian calculus
			Curve length and path energy
			Levi-Civita connection
			Parallel transport, geodesics, exponential map, Riemann curvature tensor
		Riemannian metric and Hessian structure
			Riemannian metric from Hessian
		Elastic thin shell energies
			Membrane energy
			Bending energy
			Deformation energy of thin shells
		Hessian structure on thin shells
		Hessian structure on spatially discrete shells
			Membrane energy
			Bending energy
			Hessian of discrete elastic energy induces metric
	Discrete Riemannian calculus
		Time-discrete geodesic paths
		Time-discrete geodesic calculus
	Discrete Riemannian splines
		Continuous Riemannian splines
		Time-discrete Riemannian splines
	Discrete Riemannian calculus in discrete shell space
		Interpolation in discrete shell space
			Natural regularization and physical tuning
			Lack of symmetry
			Nonuniqueness
		Extrapolation in discrete shell space
		Parallel transport in discrete shell space
		Spline interpolation in discrete shell space
			Splines in vertex space
			LΘA approximation
			Discussion
	References
Index
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