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9780824703301

Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods

by ;
  • ISBN13:

    9780824703301

  • ISBN10:

    0824703308

  • Format: Hardcover
  • Copyright: 2000-01-03
  • Publisher: CRC Press

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Summary

This text presents a comprehensive mathematical theory for elliptic, parabolic, and hyperbolic differential equations. It compares finite element and finite difference methods and illustrates applications of generalized difference methods to elastic bodies, electromagnetic fields, underground water pollution, and coupled sound-heat flows.

Author Biography

Ronghua Li is Professor of Mathematics, Head of the Department of Mathematics, and Head of the Mathematical Institute at Jilin University, Changchun, China.

Table of Contents

Preface iii
Preliminaries
1(44)
Sobolev Spaces
1(18)
Smooth approximations. Fundamental lemma of variational methods
1(2)
Generalized derivatives and Sobolev spaces
3(4)
Imbedding and trace theorems
7(2)
Finite element spaces
9(6)
Interpolation error estimates in Sobolev spaces
15(4)
Variational Problems and Their Approximations
19(26)
Abstract variational from
19(4)
Green's Formulas and variational problems
23(4)
Well-posedness of variational problems
27(2)
Approximation methods. A necessary and sufficient condition for approximate-solvability
29(6)
Galerkin methods
35(3)
Generalized Galerkin methods
38(5)
Bibliography and Comments
43(2)
Two Point Boundary Value Problems
45(190)
Basic Ideas of the Generalized Difference Method
45(8)
A Variational From
45(2)
Galerkin methods
47(1)
Generalized Galerkin Variational principles
48(4)
Generalized difference methods
52(1)
Linear Element Difference Schemes
53(8)
Trial and test function spaces
53(3)
Difference equations
56(1)
Convergence estimates
57(4)
Quadratic Element Difference Schemes
61(11)
Trial and test spaces
61(2)
Difference equations
63(3)
Convergence order estimates
66(6)
Cubic Element Difference Schemes
72(16)
Trial and test spaces
73(3)
Generalized difference methods
76(3)
Some lemmas
79(5)
Existence, uniqueness and stability
84(2)
Convergence order estimates
86(1)
Numerical examples
87(1)
Estimates in L2 and Maximum Norms
88(3)
L2-estimates
88(3)
Maximum norm estimates
91(1)
Superconvergence
91(10)
Optimal stress points
92(4)
Superconvergence for linear element difference schemes
96(1)
Superconvergence for cubic element difference schemes
97(4)
Generalized Difference Methods for a Fourth Order Equation
101(10)
Generalized difference equations
101(3)
Positive definiteness of a(uh, &Pihuh)
104(2)
Convergence order estimates
106(2)
Numerical Examples
108(1)
Bibliography and Comments
108(3)
Second Order Elliptic Equations
Introduction
111(3)
Generalized Difference Methods on Triangular Meshes
114(17)
Trial and test function spaces
114(5)
Generalized difference equation
119(5)
a priori estimates
124(5)
Error estimates
129(2)
Generalized Difference Methods on Quadrilateral Meshes
131(8)
Trial and test function spaces
131(3)
Generalized difference equation
134(3)
Convergence order estimates
137(2)
Quadratic Element Difference Schemes
139(15)
Trial and test function spaces
139(2)
Generalized difference equation
141(4)
a priori estimates
145(5)
Error estimates
150(2)
Numerical example
152(2)
Cubic Element Difference Schemes
154(13)
Trial and test function spaces
154(2)
Generalized difference equation
156(2)
a priori estimates
158(6)
Error estimates
164(3)
L2 and Maximum North Estimates
167(7)
L2 Estimates
167(6)
A maximum estimate and some remarks
173(1)
Superconvergences
174(13)
Week estimate of interpolations
174(7)
Superconvergence estimates
181(1)
Bibliography and comments
182(5)
Fourth Order and Nonlinear Elliptic Equations
Mixed generalized Difference Methods Based on Ciarlet-Raviart Variational Principle
187(9)
Mixed generalized Difference equations
188(5)
Error Estimates
193(3)
Mixed Generalized Difference Methods Based on Hermann-Miyoshi Variational Principle
196(8)
Mixed generalized difference equations
197(1)
Numerical experiments
198(6)
Nonconforming Generalized Difference Method Based on Zienkiewicz Elements
204(16)
Variational principle
202(2)
Generalized difference schemes based on Zienkiewicz elements
204(3)
Error analyses
207(11)
Numerical experiment
218(2)
Nonconforming Generalized Difference Methods Based on Adini Elements
220(7)
Generalized difference Scheme
220(2)
Error Estimate
222(4)
Numerical Example
226(1)
Second Order Nonlinear Elliptic Equations
227(8)
Generalized Difference scheme
227(4)
Error Estimate
231(2)
Bibliography and Comments
233(2)
Parabolic Equations
235(52)
Semi-discrete Generalized Difference Schemes
235(9)
Problem and schemes
235(1)
Some lemmas
235(3)
L2-error estimate
238(4)
H1-error estimate
242(2)
Fully-discrete Generalized Difference Schemes
244(10)
Fully-Discrete schemes
244(1)
Error Estimates for backward Euler generalized difference schemes
245(5)
Error estimates for Crank-Nicolson generalized difference schemes
250(4)
Mass Concentration Methods
254(7)
Construction of schemes
254(2)
Error estimates for fully-discrete schemes
256(2)
Error estimates for fully-discrete schemes
258(3)
High Order Element Difference Schemes
261(11)
Cubic element difference schemes for one dimensional papabolic equations
261(6)
Quadratic element difference schemes for two dimensional parabolic equations
267(5)
Generalized Difference Methods for Nonlinear Parabolic Equations
272(15)
Problem and Schemes
272(4)
Some lemmas
276(7)
Error estimates
283(2)
Bibliography and Comments
285(2)
Hyperbolic Equations
287(38)
Generalized difference Methods for Second Order Hyperbolic Equations
287(9)
Semi-Discrete generalized difference scheme
288(4)
Fully-discrete generalized difference scheme
292(4)
Generalized Upwind Schemes for First Order Hyperbolic Equations
296(11)
Generalized upwind schemes
297(3)
Semi-discrete error estimates
300(3)
Fully-Discrete error estimates
303(4)
Generalized Upwind Schemes for First Order Hyperbolic Systems
307(10)
Integral forms
307(2)
Generalized upwind difference schemes
309(1)
Estimation of a bilinear form
310(3)
Some practical difference schemes
313(3)
A numerical example
316(1)
Finite Volume Methods for Nonlinear Conservative Hyperbolic Equations
317(8)
Bibliography and Comments
323(2)
Convection-Dominated Diffusion Problems
325(36)
One-Dimensional Characteristic Difference Schemes
326(6)
Difference methods based on algebraic interpolations
328(2)
Upwind difference schemes
330(2)
Generalized Upwind Difference Schemes for Steady-state Problems
332(13)
Construction of the difference schemes
333(3)
Convergence and error estimate
336(3)
Extreme value theorem and uniform convergence
339(5)
Mass conservation
344(1)
Generalized Upwind Difference Schemes for Nonsteady-state Problems
345(6)
Construction of difference schemes
346(2)
Convergence and error estimate
348(3)
Highly Accurate Generalized upwind Schemes
351(6)
Construction of the difference schemes
351(4)
Convergence and error estimate
355(2)
Upwind Schemes for Nonlinear Convection Problems
357(4)
Bibliography and Comments
360(1)
Applications
361(60)
Planar Elastic Problems
361(6)
Displacement methods
362(3)
Mixed methods
365(2)
Computation of Electromagnetic Fields
367(6)
Numerical Simulation of Underground Water Pollution
373(12)
Generalized difference scheme
374(4)
Generalized upwind difference schemes
378(4)
Upwind weighted multi-element balancing method
382(3)
Stokes Equation
385(9)
Nonconforming generalized difference method
386(3)
Convergence and error estimate
389(4)
A numerical Example
393(1)
Coupled Sound-Heat Problems
394(5)
Regularized Long Wave Equations
399(10)
Semi-discrete generalized difference schemes
399(5)
Fully-discrete generalized difference schemes
404(3)
Numerical experiments
407(2)
Hierarchical Bass Methods
409(12)
Hierarchical Basis
410(3)
Application to difference equations
413(3)
Iteration methods
416(1)
Numerical experiments
417(1)
Bibliography and Comments
418(3)
Bibliography 421(18)
Index 439

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