Finite Element Methods And Their Applications

by Chen, Zhangxin

• ISBN13: 9783540240785
• ISBN10: 3540240780
• Format: Hardcover
• Copyright: 2005-08-15
• Publisher: Springer Nature

Summary

This book serves as a text for one- or two-semester courses for upper-level undergraduates and beginning graduate students and as a professional reference for people who want to solve partial differential equations (PDEs) using finite element methods. The author has attempted to introduce every concept in the simplest possible setting and maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Quite a lot of attention is given to discontinuous finite elements, characteristic finite elements, and to the applications in fluid and solid mechanics including applications to porous media flow, and applications to semiconductor modeling. An extensive set of exercises and references in each chapter are provided.

Table of Contents

1 Elementary Finite Elements	1	(86)
1.1 Introduction	2	(17)
1.1.1 A One-Dimensional Model Problem	2	(7)
1.1.2 A Two-Dimensional Model Problem	9	(5)
1.1.3 An Extension to General Boundary Conditions	14	(2)
1.1.4 Programming Considerations	16	(3)
1.2 Sobolev Spaces	19	(7)
1.2.1 Lebesgue Spaces	20	(1)
1.2.2 Weak Derivatives	21	(1)
1.2.3 Sobolev Spaces	22	(1)
1.2.4 Poincaré's Inequality	23	(2)
1.2.5 Duality and Negative Norms	25	(1)
1.3 Abstract Variational Formulation	26	(9)
1.3.1 An Abstract Formulation	26	(2)
1.3.2 The Finite Element Method	28	(2)
1.3.3 Examples	30	(5)
1.4 Finite Element Spaces	35	(11)
1.4.1 Triangles	35	(5)
1.4.2 Rectangles	40	(2)
1.4.3 Three Dimensions	42	(2)
1.4.4 A C¹ Element	44	(2)
1.5 General Domains	46	(3)
1.6 Quadrature Rules	49	(1)
1.7 Finite Elements for Transient Problems	50	(8)
1.7.1 A One-Dimensional Model Problem	51	(1)
1.7.2 A Semi-Discrete Scheme in Space	52	(3)
1.7.3 Fully Discrete Schemes	55	(3)
1.8 Finite Elements for Nonlinear Problems	58	(4)
1.8.1 Linearization Approaches	59	(1)
1.8.2 Implicit Time Approximations	60	(1)
1.8.3 Explicit Time Approximations	61	(1)
1.9 Approximation Theory	62	(8)
1.9.1 Interpolation Errors	62	(5)
1.9.2 Error Estimates for Elliptic Problems	67	(1)
1.9.3 L²-Error Estimates	68	(2)
1.10 Linear System Solution Techniques	70	(11)
1.10.1 Gaussian Elimination	70	(6)
1.10.2 The Conjugate Gradient Algorithm	76	(5)
1.11 Bibliographical Remarks	81	(1)
1.12 Exercises	81	(6)
2 Nonconforming Finite Elements	87	(30)
2.1 Second-Order Problems	87	(11)
2.1.1 Nonconforming Finite Elements on Triangles	89	(3)
2.1.2 Nonconforming Finite Elements on Rectangles	92	(3)
2.1.3 Nonconforming Finite Elements on Tetrahedra	95	(1)
2.1.4 Nonconforming Finite Elements on Parallelepipeds	95	(2)
2.1.5 Nonconforming Finite Elements on Prisms	97	(1)
2.2 Fourth-Order Problems	98	(7)
2.2.1 The Morley Element	100	(2)
2.2.2 The Fraeijs de Veubeke Element	102	(1)
2.2.3 The Zienkiewicz Element	103	(1)
2.2.4 The Adini Element	104	(1)
2.3 Nonlinear Problems	105	(1)
2.4 Theoretical Considerations	106	(7)
2.4.1 An Abstract Formulation	106	(3)
2.4.2 Applications	109	(4)
2.5 Bibliographical Remarks	113	(1)
2.6 Exercises	113	(4)
3 Mixed Finite Elements	117	(56)
3.1 A One-Dimensional Model Problem	118	(5)
3.2 A Two-Dimensional Model Problem	123	(3)
3.3 Extension to Boundary Conditions of Other Types	126	(2)
3.3.1 A Neumann Boundary Condition	126	(2)
3.3.2 A Boundary Condition of Third Type	128	(1)
3.4 Mixed Finite Element Spaces	128	(15)
3.4.1 Mixed Finite Element Spaces on Triangles	130	(3)
3.4.2 Mixed Finite Element Spaces on Rectangles	133	(3)
3.4.3 Mixed Finite Element Spaces on Tetrahedra	136	(1)
3.4.4 Mixed Finite Element Spaces on Parallelepipeds	137	(3)
3.4.5 Mixed Finite Element Spaces on Prisms	140	(3)
3.5 Approximation Properties	143	(1)
3.6 Mixed Methods for Nonlinear Problems	143	(2)
3.7 Linear System Solution Techniques	145	(9)
3.7.1 Introduction	145	(1)
3.7.2 The Uzawa Algorithm	146	(1)
3.7.3 The Minimum Residual Iterative Algorithm	147	(1)
3.7.4 Alternating Direction Iterative Algorithms	148	(2)
3.7.5 Mixed-Hybrid Algorithms	150	(2)
3.7.6 An Equivalence Relationship	152	(2)
3.8 Theoretical Considerations	154	(12)
3.8.1 An Abstract Formulation	154	(4)
3.8.2 The Mixed Finite Element Method	158	(3)
3.8.3 Examples	161	(1)
3.8.4 Construction of Projection Operators	162	(2)
3.8.5 Error Estimates	164	(2)
3.9 Bibliographical Remarks	166	(1)
3.10 Exercises	167	(6)
4 Discontinuous Finite Elements	173	(42)
4.1 Advection Problems	173	(10)
4.1.1 DG Methods	174	(4)
4.1.2 Stabilized DG Methods	178	(5)
4.2 Diffusion Problems	183	(11)
4.2.1 Symmetric DG Method	186	(1)
4.2.2 Symmetric Interior Penalty DG Method	187	(1)
4.2.3 Non-Symmetric DG Method	188	(1)
4.2.4 Non-Symmetric Interior Penalty DG Method	189	(3)
4.2.5 Remarks	192	(2)
4.3 Mixed Discontinuous Finite Elements	194	(14)
4.3.1 A One-Dimensional Problem	194	(9)
4.3.2 Multi-Dimensional Problems	203	(3)
4.3.3 Nonlinear Problems	206	(2)
4.4 Theoretical Considerations	208	(4)
4.4.1 DG Methods	208	(2)
4.4.2 Stabilized DG Methods	210	(2)
4.5 Bibliographical Remarks	212	(1)
4.6 Exercises	212	(3)
5 Characteristic Finite Elements	215	(46)
5.1 An Example	216	(2)
5.2 The Modified Method of Characteristics	218	(8)
5.2.1 A One-Dimensional Model Problem	218	(4)
5.2.2 Periodic Boundary Conditions	222	(1)
5.2.3 Extension to Multi-Dimensional Problems	222	(2)
5.2.4 Discussion of a Conservation Relation	224	(2)
5.3 The Eulerian-Lagrangian Localized Adjoint Method	226	(16)
5.3.1 A One-Dimensional Model Problem	226	(10)
5.3.2 Extension to Multi-Dimensional Problems	236	(6)
5.4 The Characteristic Mixed Method	242	(3)
5.5 The Eulerian-Lagrangian Mixed Discontinuous Method	245	(3)
5.6 Nonlinear Problems	248	(2)
5.7 Remarks on Characteristic Finite Elements	250	(1)
5.8 Theoretical Considerations	250	(8)
5.9 Bibliographical Remarks	258	(1)
5.10 Exercises	258	(3)
6 Adaptive Finite Elements	261	(44)
6.1 Local Grid Refinement in Space	262	(5)
6.1.1 Regular H-Schemes	263	(2)
6.1.2 Irregular H-Schemes	265	(1)
6.1.3 Unrefinements	266	(1)
6.2 Data Structures	267	(3)
6.3 A-Posteriori Error Estimates for Stationary Problems	270	(19)
6.3.1 Residual Estimators	271	(6)
6.3.2 Local Problem-Based Estimators	277	(4)
6.3.3 Averaging-Based Estimators	281	(2)
6.3.4 Hierarchical Basis Estimators	283	(4)
6.3.5 Efficiency of Error Estimators	287	(2)
6.4 A-Posteriori Error Estimates for Transient Problems	289	(3)
6.5 A-Posteriori Error Estimates for Nonlinear Problems	292	(1)
6.6 Theoretical Considerations	293	(9)
6.6.1 An Abstract Theory	294	(3)
6.6.2 Applications	297	(5)
6.7 Bibliographical Remarks	302	(1)
6.8 Exercises	302	(3)
7 Solid Mechanics	305	(16)
7.1 Introduction	305	(3)
7.1.1 Kinematics	305	(1)
7.1.2 Equilibrium	306	(1)
7.1.3 Material Laws	306	(2)
7.2 Variational Formulations	308	(2)
7.2.1 The Displacement Form	308	(1)
7.2.2 The Mixed Form	309	(1)
7.3 Finite Element Methods	310	(4)
7.3.1 Finite Elements and Locking Effects	310	(1)
7.3.2 Mixed Finite Elements	311	(2)
7.3.3 Nonconforming Finite Elements	313	(1)
7.4 Theoretical Considerations	314	(5)
7.5 Bibliographical Remarks	319	(1)
7.6 Exercises	319	(2)
8 Fluid Mechanics	321	(16)
8.1 Introduction	321	(2)
8.2 Variational Formulations	323	(1)
8.2.1 The Galerkin Approach	323	(1)
8.2.2 The Mixed Formulation	324	(1)
8.3 Finite Element Methods	324	(5)
8.3.1 Galerkin Finite Elements	324	(1)
8.3.2 Mixed Finite Elements	325	(1)
8.3.3 Nonconforming Finite Elements	326	(3)
8.4 The Navier-Stokes Equation	329	(1)
8.5 Theoretical Considerations	330	(3)
8.6 Bibliographical Remarks	333	(1)
8.7 Exercises	333	(4)
9 Fluid Flow in Porous Media	337	(26)
9.1 Two-Phase Immiscible Flow	338	(5)
9.1.1 The Phase Formulation	340	(2)
9.1.2 The Weighted Formulation	342	(1)
9.1.3 The Global Formulation	342	(1)
9.2 Mixed Finite Elements for Pressure	343	(2)
9.3 Characteristic Methods for Saturation	345	(1)
9.4 A Numerical Example	346	(3)
9.5 Theoretical Considerations	349	(12)
9.5.1 Analysis for the Pressure Equation	349	(2)
9.5.2 Analysis for the Saturation Equation	351	(10)
9.6 Bibliographical Remarks	361	(1)
9.7 Exercises	362	(1)
10 Semiconductor Modeling	363	(22)
10.1 Three Semiconductor Models	364	(4)
10.1.1 The Drift-Diffusion Model	364	(2)
10.1.2 The Hydrodynamic Model	366	(1)
10.1.3 The Quantum Hydrodynamic Model	367	(1)
10.2 Numerical Methods	368	(11)
10.2.1 The Drift-Diffusion Model	368	(3)
10.2.2 The Hydrodynamic Model	371	(7)
10.2.3 The Quantum Hydrodynamic Model	378	(1)
10.3 A Numerical Example	379	(5)
10.4 Bibliographical Remarks	384	(1)
10.5 Exercises	384	(1)
A Nomenclature	385	(6)
References	391	(14)
Index	405

Excerpts

"This book serves as a text for one- or two-semester courses for upper-level undergraduates and beginning graduate students and as a professional reference for people who want to solve partial differential equations (PDEs) using finite element methods. The author has attempted to introduce every concept in the simplest possible setting and maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Quite a lot of attention is given to discontinuous finite elements, characteristic finite elements, and to the applications in fluid and solid mechanics including applications to porous media flow, and applications to semiconductor modeling. An extensive set of exercises and references in each chapter are provided."--BOOK JACKET.