Abstract: Publication date: Available online 11 November 2020Source: Handbook of Numerical AnalysisAuthor(s): Jose A. Carrillo, Daniel Matthes, Marie-Therese Wolfram

Abstract: Publication date: Available online 31 October 2020Source: Handbook of Numerical AnalysisAuthor(s): Grégoire Allaire, Charles Dapogny, François Jouve

Abstract: Publication date: Available online 28 October 2020Source: Handbook of Numerical AnalysisAuthor(s): Thomas J.R. Hughes, Giancarlo Sangalli, Thomas Takacs, Deepesh Toshniwal

Abstract: Publication date: Available online 19 December 2019Source: Handbook of Numerical AnalysisAuthor(s): John W. Barrett, Harald Garcke, Robert NürnbergAbstractParametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very nonuniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.

Abstract: Publication date: Available online 18 October 2019Source: Handbook of Numerical AnalysisAuthor(s): Robert I. Saye, James A. SethianAbstractLevel set methods, and their descendents, have been valuable in providing robust numerical algorithms for tracking the dynamics of evolving interfaces moving under complex physics and have been instrumental in shedding light on a variety of scientific problems. Application areas include fluid mechanics, materials sciences, biology, and chemistry, as well as in engineering and industry such as semiconductor manufacturing, the design of commercial inkjet plotters, and the dynamics of industrial foams. To handle increasingly complex and intricate interface dynamics problems, a set of mathematical, algorithmic, and numerical challenges had to be overcome in order to extend the basic level set methodology, introduced in Osher and Sethian (1988). In this review article, we discuss how each of those challenges were tackled and show examples of how the resulting advances in core methodologies have been used to compute a variety of complex problems in computational physics.

Abstract: Publication date: Available online 12 October 2019Source: Handbook of Numerical AnalysisAuthor(s): Behrend Heeren, Martin Rumpf, Max Wardetzky, Benedikt WirthAbstractThis article discusses a Riemannian calculus on the space of discrete shells, represented by triangular meshes, and an effective as well as efficient discretization thereof. This calculus comprises shape morphing via geodesic curves, shape extrapolation via the Riemannian exponential map, the transfer of geometric detail via Riemannian parallel transport, and smooth interpolation of multiple key frames using Riemannian splines. The underlying Riemannian structure is derived from the Hessian of an elastic deformation energy. The discretization is based on an approximation of the Riemannian path energy via a sum of deformation energies defined on pairs of triangular shapes along a finite sequence of discrete shells. Furthermore, convergence results of all components of the discrete calculus to their continuous counterparts are provided, and a variety of surface processing applications are discussed.

Abstract: Publication date: Available online 13 September 2019Source: Handbook of Numerical AnalysisAuthor(s): Andrea Bonito, Alan Demlow, Ricardo H. NochettoAbstractPartial differential equations posed on surfaces arise in a number of applications. In this survey we describe three popular finite element methods for approximating solutions to the Laplace–Beltrami problem posed on an n-dimensional surface γ embedded in Rn+1: the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface Γ whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface Γ in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface γ is embedded in an n + 1-dimensional domain Ω which has itself been triangulated. An n-dimensional approximate surface Γ is then constructed roughly speaking by interpolating γ over the triangulation of Ω, and the finite element space over Γ consists of the trace (restriction) of a standard finite element space on Ω to Γ. In the narrow band method the PDE posed on the surface is extended to a triangulated n + 1-dimensional band about γ whose width is proportional to the diameter of elements in the triangulation. In all cases we provide optimal a priori error estimates for the lowest order finite element methods, and we also present a posteriori error estimates for the parametric and trace methods. Our presentation focuses especially on the relationship between the regularity of the surface γ, which is never assumed better than of class C2, the manner in which γ is represented in theory and practice, and the properties of the resulting methods.

Abstract: Publication date: Available online 5 September 2019Source: Handbook of Numerical AnalysisAuthor(s): Qiang Du, Xiaobing FengAbstractThis chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modelling and their numerical methods in materials science, fluid mechanics, biology and image science.

Abstract: Publication date: Available online 26 July 2019Source: Handbook of Numerical AnalysisAuthor(s): Michael Neilan, Abner J. Salgado, Wujun ZhangAbstractWe review recent advances in the numerical analysis of the Monge–Ampère equation. Various computational techniques are discussed including wide stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.

Abstract: Publication date: Available online 24 July 2019Source: Handbook of Numerical AnalysisAuthor(s): Sören BartelsAbstractNonlinear bending phenomena of thin elastic structures arise in various modern and classical applications. Characterizing low energy states of elastic rods has been investigated by Bernoulli in 1738 and related models are used to determine configurations of DNA strands. The bending of a piece of paper has been described mathematically by Kirchhoff in 1850 and extensions of his model arise in nanotechnological applications such as the development of externally operated microtools. A rigorous mathematical framework that identifies these models as dimensionally reduced limits from three-dimensional hyperelasticity has only recently been established. It provides a solid basis for developing and analyzing numerical approximation schemes. The fourth-order character of bending problems and a pointwise isometry constraint for large deformations require appropriate discretization techniques which are discussed in this article. Methods developed for the approximation of harmonic maps are adapted to discretize the isometry constraint and gradient flows are used to decrease the bending energy. For the case of elastic rods, torsion effects and a self-avoidance potential that guarantees injectivity of deformations are incorporated. The devised and rigorously analyzed numerical methods are illustrated by means of experiments related to the relaxation of elastic knots, the formation of singularities in a Möbius strip, and the simulation of actuated bilayer plates.

Abstract: Publication date: Available online 2 July 2019Source: Handbook of Numerical AnalysisAuthor(s): Eberhard Bänsch, Alfred SchmidtAbstractFree boundary problems with geometric conditions for the interfaces are typically models for situations where interfacial energies play an important role. This happens for example in microscale models of phase transitions as well as models with fluidic interfaces in various scales, all important in technical applications. We give here an overview of related aspects of models and corresponding numerical methods, and present some representative applications and results.