High-Order Finite Difference and Finite Element Methods for Solving Some Partial Differential Equations (Synthesis Lectures on Engineering, Science, and Technology)

Preface
Contents
List of 
Figures
List of 
Tables
1 Accurate Finite-Difference Methods for Helmholtz and Wave Equations
	1.1 Introduction
	1.2 Accurate Finite Difference Methods for the Helmholtz Equation
		1.2.1 Statement of the Problem
		1.2.2 Construction of the Accurate Finite Difference Equations
		1.2.3 Accurate Finite Difference Boundary Conditions
		1.2.4 Method for Solving the Finite Difference Equations
		1.2.5 Numerical Results
	1.3 Accurate Finite Difference Method for the Wave Equation
		1.3.1 Construction of the Accurate Finite Difference Method
		1.3.2 Accurate Finite Difference Initial and Boundary Conditions
		1.3.3 Method for Solving the Finite Difference Equations
		1.3.4 Numerical Results
	1.4 Statement of Problem and Exact Finite Difference Method …
		1.4.1 Solutions of Discrete Equations and Calculation Techniques
		1.4.2 Numerical Results
2 Higher-Order Finite-Difference Methods for the Burgers\' Equations
	2.1 Introduction
	2.2 High-Order Numerical Solution of the One-Dimensional Burgers\' Equation
		2.2.1 Construction of High-Order Finite-Difference Methods
		2.2.2 Numerical Results
	2.3 High-Order Numerical Solution of the Unsteady Burgers\' Equation
		2.3.1 Reduction of the Unsteady Burgers\' Equation to the One-Dimensional Burgers\' Equation
		2.3.2 Numerical Solution of the Heat Equation
		2.3.3 High-Order Finite Difference Methods for Solution of One-Dimensional Burgers\' Equation
		2.3.4 Numerical Results
	2.4 High-Order Numerical Solution of the Two-Dimensional …
		2.4.1 The Fourth-Order Explicit Finite Difference Method
		2.4.2 Numerical Results
3 High-Accuracy Finite Element Methods for Solution of Discrete Spectrum Problems
	3.1 Introduction
	3.2 Setting of the Problem
	3.3 The Shape Functions
		3.3.1 One Dimensional Lagrange and Hermite Interpolation Polynomials
		3.3.2 Lagrange Interpolation Polynomials on Simplex
		3.3.3 The Economical Implementation of Finite Element Method
		3.3.4 Hermite Interpolation Polynomials on Simplex
		3.3.5 Hermite Interpolation Polynomials on Parallelepiped
		3.3.6 Piecewise Polynomial Functions
	3.4 Examples
		3.4.1 The Helmholtz Problem on Triangle
		3.4.2 The Helmholtz Problem on Hypercube
		3.4.3 Quadrupole-Octupole-Vibrational Collective Model
A Continuous Analogue of Newton\'s Method for Solving the Generalized Eigenvalue Problem
PI-Type Fully Symmetric Quadrature Rules on the Simplexes
	B.1  Construction of Fully Symmetric Quadrature Rules
	B.2 Estimates of the Errors of the Quadrature Rules