Geometric Partial Differential Equations - Part I

Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering.

About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimizationEach chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further readingThe authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs

1. Finite element methods for the Laplace-Beltrami operator Andrea Bonito, Alan Demlow and Ricardo H. Nochetto 2. The Monge–Ampère equation Michael Neilan, Abner J. Salgado and Wujun Zhang 3. Finite element simulation of nonlinear bending models for thin elastic rods and plates Sören Bartels 4. Parametric finite element approximations of curvature-driven interface evolutions John W. Barrett, Harald Garcke and Robert Nürnberg 5. The phase field method for geometric moving interfaces and their numerical approximations Qiang Du and Xiaobing Feng 6. A review of level set methods to model interfaces moving under complex physics: Recent challenges and advances Robert I. Saye and James A. Sethian 7. Free boundary problems in fluids and materials Eberhard Bänsch and Alfred Schmidt 8. Discrete Riemannian calculus on shell space Behrend Heeren, Martin Rumpf, Max Wardetzky and Benedikt Wirth