Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, creating an exciting interplay between the subjects. This is the first and only book to prove in a systematic and unifying way, stability, convergence and computing results for the different numerical methods for nonlinear elliptic problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. Examples are given for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference, wavelet (and, in a volume to follow, spectral and meshfree) methods. A number of specific long open problems are solved here: numerical methods for fully nonlinear elliptic problems, wavelet and meshfree methods for nonlinear problems, and more general nonlinear boundary conditions. We apply it to all these problems and methods, in particular to eigenvalues, monotone operators, quadrature approximations, and Newton methods. Adaptivity is discussed for finite element and wavelet methods. The book has been written for graduate students and scientists who want to study and to numerically analyze nonlinear elliptic differential equations in Mathematics, Science and Engineering. It can be used as material for graduate courses or advanced seminars.
Author(s): Klaus Bohmer
Series: Numerical Mathematics and Scientific Computation
Publisher: Oxford University Press, USA
Year: 2010
Language: English
Pages: 775
Tags: Математика;Вычислительная математика;
Contents......Page 8
Preface......Page 15
PART I: ANALYTICAL RESULTS......Page 30
1.2 Linear versus nonlinear models......Page 32
1.3 Examples for nonlinear partial differential equations......Page 39
1.4 Fundamental results......Page 42
2.1 Introduction......Page 61
2.2 Linear elliptic differential operators of second order, bilinear forms and solution concepts......Page 65
2.3 Bilinear forms and induced linear operators......Page 74
2.4 Linear elliptic differential operators, Fredholm alternative and regular solutions......Page 83
2.5 Nonlinear elliptic equations......Page 106
2.6 Linear and nonlinear elliptic systems......Page 142
2.7 Linearization of nonlinear operators......Page 176
2.8 The Navier–Stokes equation......Page 192
PART II: NUMERICAL METHODS......Page 200
3.1 Introduction......Page 202
3.2 Petrov–Galerkin and general discretization methods......Page 204
3.3 Variational and classical consistency......Page 214
3.4 Stability and consistency yield convergence......Page 218
3.5 Techniques for proving stability......Page 223
3.6 Stability implies invertibility......Page 232
3.7 Solving nonlinear systems: Continuation and Newton’s method based upon the mesh independence principle (MIP)......Page 234
4.1 Introduction......Page 238
4.2 Approximation theory for finite elements......Page 241
4.3 FEMs for linear problems......Page 286
4.4 Finite element methods for divergent quasilinear elliptic equations and systems......Page 302
4.5 General convergence theory for monotone and quasilinear operators......Page 306
4.6 Mixed FEMs for Navier–Stokes and saddle point equations......Page 310
4.7 Variational methods for eigenvalue problems......Page 317
5.1 Introduction......Page 325
5.2 Finite element methods for fully nonlinear elliptic problems......Page 327
5.3 FE and other methods for nonlinear boundary conditions......Page 374
5.4 Quadrature approximate FEMs......Page 375
5.5 Consistency, stability and convergence for FEMs with variational crimes......Page 397
6.1 Introduction......Page 449
6.2 The residual error estimator for the Poisson problem......Page 459
6.3 Estimation of quantities of interest......Page 478
7.1 Introduction......Page 484
7.2 The model problem......Page 488
7.3 Discretization of the problem......Page 490
7.4 General linear elliptic problems......Page 501
7.5 Semilinear and quasilinear elliptic problems......Page 503
7.6 DCGMs are general discretization methods......Page 511
7.7 Geometry of the mesh, error and inverse estimates......Page 515
7.8 Penalty norms and consistency of the Jσ[sub(h)]......Page 520
7.9 Coercive linearized principal parts......Page 523
7.10 Consistency results for the c[sup(h)], b[sup(h)], l[sup(h)]......Page 532
7.11 Consistency properties of the a[sub(h)]......Page 536
7.12 Convergence for DCGMs......Page 556
7.13 Solving nonlinear equations in DCGMs......Page 561
7.14 hp-variants of DCGM......Page 567
7.15 Numerical experiences......Page 575
8.1 Introduction......Page 589
8.2 Difference methods for simple examples, notation......Page 591
8.3 Discrete Sobolev spaces......Page 595
8.4 General elliptic problems with Dirichlet conditions, and their difference methods......Page 601
8.5 Convergence for difference methods......Page 617
8.6 Natural boundary value problems of order 2......Page 639
8.7 Other difference methods on curved boundaries......Page 651
8.8 Asymptotic expansions, extrapolation, and defect corrections......Page 655
8.9 Numerical experiments for the von Kármán equations......Page 662
9.1 Introduction......Page 664
9.2 The scope of problems......Page 666
9.3 Wavelet analysis......Page 668
9.4 Stable discretizations and preconditioning......Page 682
9.5 Applications to elliptic equations......Page 688
9.6 Saddle point and (Navier–)Stokes equations......Page 693
9.7 Adaptive wavelet methods......Page 698
Bibliography......Page 715
Index......Page 762
B......Page 763
C......Page 764
D......Page 765
E......Page 766
F......Page 768
I......Page 769
M......Page 770
O......Page 771
P......Page 772
S......Page 773
T......Page 774
W......Page 775
