Approximation of Elliptic Boundary-Value Problems
Title | Approximation of Elliptic Boundary-Value Problems PDF eBook |
Author | Jean-Pierre Aubin |
Publisher | Courier Corporation |
Pages | 386 |
Release | 2007-01-01 |
Genre | Mathematics |
ISBN | 0486457915 |
A marriage of the finite-differences method with variational methods for solving boundary-value problems, the finite-element method is superior in many ways to finite-differences alone. This self-contained text for advanced undergraduates and graduate students is intended to imbed this combination of methods into the framework of functional analysis and to explain its applications to approximation of nonhomogeneous boundary-value problems for elliptic operators. The treatment begins with a summary of the main results established in the book. Chapter 1 introduces the variational method and the finite-difference method in the simple case of second-order differential equations. Chapters 2 and 3 concern abstract approximations of Hilbert spaces and linear operators, and Chapters 4 and 5 study finite-element approximations of Sobolev spaces. The remaining four chapters consider several methods for approximating nonhomogeneous boundary-value problems for elliptic operators.
Numerical Approximation Methods for Elliptic Boundary Value Problems
Title | Numerical Approximation Methods for Elliptic Boundary Value Problems PDF eBook |
Author | Olaf Steinbach |
Publisher | Springer Science & Business Media |
Pages | 392 |
Release | 2007-12-22 |
Genre | Mathematics |
ISBN | 0387688056 |
This book presents a unified theory of the Finite Element Method and the Boundary Element Method for a numerical solution of second order elliptic boundary value problems. This includes the solvability, stability, and error analysis as well as efficient methods to solve the resulting linear systems. Applications are the potential equation, the system of linear elastostatics and the Stokes system. While there are textbooks on the finite element method, this is one of the first books on Theory of Boundary Element Methods. It is suitable for self study and exercises are included.
On the Approximation of Elliptic Boundary Value Problems by Fundamental Solutions
Title | On the Approximation of Elliptic Boundary Value Problems by Fundamental Solutions PDF eBook |
Author | Mathon, R |
Publisher | University of Toronto, Department of Computer Science |
Pages | 256 |
Release | 1972 |
Genre | |
ISBN |
Hybrid finite element approximation of elliptic boundary value problems
Title | Hybrid finite element approximation of elliptic boundary value problems PDF eBook |
Author | Alfio Quarteroni |
Publisher | |
Pages | 45 |
Release | 1979 |
Genre | |
ISBN |
Optimization in Solving Elliptic Problems
Title | Optimization in Solving Elliptic Problems PDF eBook |
Author | Eugene G. D'yakonov |
Publisher | CRC Press |
Pages | 590 |
Release | 2018-05-04 |
Genre | Mathematics |
ISBN | 135108366X |
Optimization in Solving Elliptic Problems focuses on one of the most interesting and challenging problems of computational mathematics - the optimization of numerical algorithms for solving elliptic problems. It presents detailed discussions of how asymptotically optimal algorithms may be applied to elliptic problems to obtain numerical solutions meeting certain specified requirements. Beginning with an outline of the fundamental principles of numerical methods, this book describes how to construct special modifications of classical finite element methods such that for the arising grid systems, asymptotically optimal iterative methods can be applied. Optimization in Solving Elliptic Problems describes the construction of computational algorithms resulting in the required accuracy of a solution and having a pre-determined computational complexity. Construction of asymptotically optimal algorithms is demonstrated for multi-dimensional elliptic boundary value problems under general conditions. In addition, algorithms are developed for eigenvalue problems and Navier-Stokes problems. The development of these algorithms is based on detailed discussions of topics that include accuracy estimates of projective and difference methods, topologically equivalent grids and triangulations, general theorems on convergence of iterative methods, mixed finite element methods for Stokes-type problems, methods of solving fourth-order problems, and methods for solving classical elasticity problems. Furthermore, the text provides methods for managing basic iterative methods such as domain decomposition and multigrid methods. These methods, clearly developed and explained in the text, may be used to develop algorithms for solving applied elliptic problems. The mathematics necessary to understand the development of such algorithms is provided in the introductory material within the text, and common specifications of algorithms that have been developed for typical problems in mathema
Elliptic Problems in Nonsmooth Domains
Title | Elliptic Problems in Nonsmooth Domains PDF eBook |
Author | Pierre Grisvard |
Publisher | SIAM |
Pages | 426 |
Release | 2011-10-20 |
Genre | Mathematics |
ISBN | 1611972027 |
Originally published: Boston: Pitman Advanced Pub. Program, 1985.
Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation
Title | Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation PDF eBook |
Author | Zohar Yosibash |
Publisher | Springer Science & Business Media |
Pages | 473 |
Release | 2011-12-02 |
Genre | Mathematics |
ISBN | 146141508X |
This introductory and self-contained book gathers as much explicit mathematical results on the linear-elastic and heat-conduction solutions in the neighborhood of singular points in two-dimensional domains, and singular edges and vertices in three-dimensional domains. These are presented in an engineering terminology for practical usage. The author treats the mathematical formulations from an engineering viewpoint and presents high-order finite-element methods for the computation of singular solutions in isotropic and anisotropic materials, and multi-material interfaces. The proper interpretation of the results in engineering practice is advocated, so that the computed data can be correlated to experimental observations. The book is divided into fourteen chapters, each containing several sections. Most of it (the first nine Chapters) addresses two-dimensional domains, where only singular points exist. The solution in a vicinity of these points admits an asymptotic expansion composed of eigenpairs and associated generalized flux/stress intensity factors (GFIFs/GSIFs), which are being computed analytically when possible or by finite element methods otherwise. Singular points associated with weakly coupled thermoelasticity in the vicinity of singularities are also addressed and thermal GSIFs are computed. The computed data is important in engineering practice for predicting failure initiation in brittle material on a daily basis. Several failure laws for two-dimensional domains with V-notches are presented and their validity is examined by comparison to experimental observations. A sufficient simple and reliable condition for predicting failure initiation (crack formation) in micron level electronic devices, involving singular points, is still a topic of active research and interest, and is addressed herein. Explicit singular solutions in the vicinity of vertices and edges in three-dimensional domains are provided in the remaining five chapters. New methods for the computation of generalized edge flux/stress intensity functions along singular edges are presented and demonstrated by several example problems from the field of fracture mechanics; including anisotropic domains and bimaterial interfaces. Circular edges are also presented and the author concludes with some remarks on open questions. This well illustrated book will appeal to both applied mathematicians and engineers working in the field of fracture mechanics and singularities.