Numerical Solution of Partial Differential Equations by the Finite Element Method

ISBN-10: 048646900X
ISBN-13: 9780486469003
Edition: 2009
Authors: Claes Johnson
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Description: An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear  More...

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Book details

List price: $18.95
Copyright year: 2009
Publisher: Dover Publications, Incorporated
Publication date: 1/15/2009
Binding: Paperback
Pages: 288
Size: 6.00" wide x 9.00" long x 0.50" tall
Weight: 0.792
Language: English

An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science-and engineering-related specialties.

Preface to the Dover Edition
Preface
Introduction
Background
Difference methods - Finite element methods
Scope of the book
Introduction to FEM for elliptic problems
Variational formulation of a one-dimensional model problem
FEM for the model problem with piecewise linear functions
An error estimate for FEM for the model problem
FEM for the Poisson equation
The Hilbert spaces L2(�), H1 (�) and H10(�)
A geometric interpretation of FEM
A Neumann problem. Natural and essential boundary conditions
Remarks on programming
Remarks on finite element software
Abstract formulation of the finite element method for elliptic problems
Introduction. The continuous problem
Discretization. An error estimate
The energy norm
Some examples
Some finite element spaces
Introduction. Regularity requirements
Some examples of finite elements
Summary
Approximation theory for FEM. Error estimates for elliptic problems
Introduction
Interpolation with piecewise linear functions in two dimensions
Interpolation with polynomials of higher degree
Error estimates for FEM for elliptic problems
On the regularity of the exact solution
Adaptive methods
An error estimate in the L2(�)-nom
Some applications to elliptic problems
The elasticity problem
Stokes problem
A plate problem
Direct methods for solving linear systems of equations
Introduction
Gaussian elimination. Cholesky's method
Operation counts. Band matrices
Fill-in
The frontal method
Nested dissection
Minimization algorithms. Iterative methods
Introduction
The gradient method
The conjugate gradient method
Preconditioning
Multigrid methods
Work estimates for direct and iterative methods
The condition number of the stiffness matrix
FEM for parabolic problems
Introduction
A one-dimensional model problem
Semi-discretization in space
Discretization in space and time
Background
The backward Euler and Crank-Nicolson methods
The discontinuous Galerkin method
Error estimates for fully discrete approximations and automatic time and space step control
Hyperbolic problems
Introduction
A convection-diffusion problem
General remarks on numerical methods for hyperbolic equations
Outline and preliminaries
Standard Galerkin
Classical artificial diffusion
The streamline diffusion method
The streamline diffusion method with �=0
The streamline diffusion method with �>0
The discontinuous Galerkin method
The streamline diffusion method for time-dependent convection-diffusion problems
Friedrichs' systems
The continuous problem
The standard Galerkin method
The streamline diffusion method
The discontinuous Galerkin method
Second order hyperbolic problems
Boundary element methods
Introduction
Some integral equations
An integral equation for an exterior Dirichlet problem using a single layer potential
An exterior Dirichlet problem with double layer potential
An exterior Neumann problem with single layer potential
Alternative integral equation formulations
Finite element methods
FEM for a Fredholm equation of the first kind
FEM for a Fredholm equation of the second kind
Mixed finite element methods
Introduction
Some examples
Curved elements and numerical integration
Curved elements
Numerical integration (quadrature)
Some non-linear problems
Introduction
Convex minimization problems
The continuous problem
Discretizations
Numerical methods for convex minimization problems
A non-linear parabolic problem
The incompressible Euler equations
The continuous problem
The streamline diffusion method in (�, �)-formulation
The discontinuous Galerkin method in (�, �)-formulation
The streamline diffusion method in (u, p)-formulation
The incompressible Navier-Stokes equations
Compressible flow: Burgers' equation
References
Index

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