Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics

by Elman, Howard C.; Silvester, David J.; Wathen, Andrew J.

• ISBN13: 9780198528685
• ISBN10: 019852868X
• Format: Paperback
• Copyright: 2005-08-04
• Publisher: Oxford University Press

Summary

The subject of this book is the efficient solution of partial differential equations (PDEs) that arise when modelling incompressible fluid flow.The material is organized into four groups of two chapters each, covering the Poisson equation (chapters 1 and 2); the convection-diffusion equation(chapters 3 and 4); the Stokes equations (chapters 5 and 6); and the Navier-Stokes equations (chapters 7 and 8). These equations represent important models within the domain of computational fluid dynamics, but they also arise in many other settings. For each PDE model, there is a chapter concernedwith finite element discretization, and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. For each problem and associated solvers there is a description of how to compute along with theoretical analysis which guides the choice ofapproaches. Illustrative numerical results occur throughout the book, which have been computed with the freely downloadable IFISS software. All numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope fopr experimentation in the 'computationallaboratory' provided by the software.This book provides an excellent introduction to finite elements, iterative linear solvers and scientific computing aimed at graduates in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. Including theoretical problems and practical exercises closelytied with freely downloadable MATLAB software, it is an ideal teaching and learning resource.

Table of Contents

0 Models of incompressible fluid flow	1	(9)
1 The Poisson equation	10	(58)
1.1 Reference problems	11	(3)
1.2 Weak formulation	14	(3)
1.3 The Galerkin finite element method	17	(10)
1.3.1 Triangular finite elements (R²)	20	(2)
1.3.2 Quadrilateral elements (R²)	22	(3)
1.3.3 Tetrahedral elements (R³)	25	(1)
1.3.4 Brick elements (R³)	26	(1)
1.4 Implementation aspects	27	(9)
1.4.1 Triangular element matrices	28	(3)
1.4.2 Quadrilateral element matrices	31	(2)
1.4.3 Assembly of the Galerkin system	33	(3)
1.5 Theory of errors	36	(20)
1.5.1 A priori error bounds	38	(10)
1.5.2 A posteriori error bounds	48	(8)
1.6 Matrix properties	56	(5)
Problems	61	(5)
Computational exercises	66	(2)
2 Solution of discrete Poisson problems	68	(45)
2.1 The conjugate gradient method	69	(9)
2.1.1 Convergence analysis	73	(2)
2.1.2 Stopping criteria	75	(3)
2.2 Preconditioning	78	(5)
2.3 Singular systems are not a problem	83	(1)
2.4 The Lanczos and minimum residual methods	84	(4)
2.5 Multigrid	88	(19)
2.5.1 Two-grid convergence theory	95	(6)
2.5.2 Extending two-grid to multigrid	101	(6)
Problems	107	(3)
Computational exercises	110	(3)
3 The convection-diffusion equation	113	(53)
3.1 Reference problems	115	(5)
3.2 Weak formulation and the convection term	120	(3)
3.3 Approximation by finite elements	123	(11)
3.3.1 The Galerkin finite element method	123	(3)
3.3.2 The streamline diffusion method	126	(8)
3.4 Theory of errors	134	(14)
3.4.1 A priori error bounds	134	(8)
3.4.2 A posteriori error bounds	142	(6)
3.5 Matrix properties	148	(13)
3.5.1 Computational molecules and Fourier analysis	152	(4)
3.5.2 Analysis of difference equations	156	(5)
Discussion and bibliographical notes	161	(2)
Problems	163	(1)
Computational exercises	164	(2)
4 Solution of discrete convection-diffusion problems	166	(48)
4.1 Krylov subspace methods	166	(10)
4.1.1 GMRES	167	(5)
4.1.2 Biorthogonalization methods	172	(4)
4.2 Preconditioning methods and splitting operators	176	(18)
4.2.1 Splitting operators for convection-diffusion systems	178	(3)
4.2.2 Matrix analysis of convergence	181	(4)
4.2.3 Asymptotic analysis of convergence	185	(5)
4.2.4 Practical considerations	190	(4)
4.3 Multigrid	194	(14)
4.3.1 Practical issues	195	(5)
4.3.2 Tools of analysis: smoothing and approximation properties	200	(2)
4.3.3 Smoothing	202	(3)
4.3.4 Analysis	205	(3)
Discussion and bibliographical notes	208	(3)
Problems	211	(1)
Computational exercises	212	(2)
5 The Stokes equations	214	(71)
5.1 Reference problems	217	(5)
5.2 Weak formulation	222	(2)
5.3 Approximation using mixed finite elements	224	(25)
5.3.1 Stable rectangular elements (Q2-Q1, Q2-P-1, Q2-P0)	229	(6)
5.3.2 Stabilized rectangular elements (Q1-P0, Q1-Q1)	235	(10)
5.3.3 Triangular elements	245	(3)
5.3.4 Brick and tetrahedral elements	248	(1)
5.4 Theory of errors	249	(19)
5.4.1 A priori error bounds	250	(12)
5.4.2 A posteriori error bounds	262	(6)
5.5 Matrix properties	268	(9)
5.5.1 Stable mixed approximation	270	(3)
5.5.2 Stabilized mixed approximation	273	(4)
Discussion and bibliographical notes	277	(3)
Problems	280	(3)
Computational exercises	283	(2)
6 Solution of discrete Stokes problems	285	(28)
6.1 The preconditioned MINRES method	286	(3)
6.2 Preconditioning	289	(19)
6.2.1 General strategies for preconditioning	291	(5)
6.2.2 Eigenvalue bounds	296	(7)
6.2.3 Equivalent norms for MINRES	303	(3)
6.2.4 MINRES convergence analysis	306	(2)
Discussion and bibliographical notes	308	(1)
Problems	309	(1)
Computational exercises	310	(3)
7 The Navier-Stokes equations	313	(28)
7.1 Reference problems	315	(3)
7.2 Weak formulation and linearization	318	(9)
7.2.1 Stability theory and bifurcation analysis	320	(4)
7.2.2 Nonlinear iteration	324	(3)
7.3 Mixed finite element approximation	327	(3)
7.4 Theory of errors	330	(7)
7.4.1 A priori error bounds	331	(2)
7.4.2 A posteriori error bounds	333	(4)
Discussion and bibliographical notes	337	(2)
Problems	339	(1)
Computational exercises	339	(2)
8 Solution of discrete Navier-Stokes problems	341	(41)
8.1 General strategies for preconditioning	342	(4)
8.2 Approximations to the Schur complement operator	346	(8)
8.2.1 The pressure convection-diffusion preconditioner	347	(6)
8.2.2 The least-squares commutator preconditioner	353	(1)
8.3 Performance and analysis	354	(16)
8.3.1 Ideal versions of the preconditioners	355	(4)
8.3.2 Use of iterative methods for subproblems	359	(5)
8.3.3 Convergence analysis	364	(1)
8.3.4 Enclosed flow: singular systems are not a problem	365	(3)
8.3.5 Relation to SIMPLE iteration	368	(2)
8.4 Nonlinear iteration	370	(5)
Discussion and bibliographical notes	375	(3)
Problems	378	(1)
Computational exercises	379	(3)
Bibliography	382	(15)
Index	397