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E-grāmata: Numerical Methods for Nonlinear Partial Differential Equations

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Numerical Methods for Nonlinear Partial Differential EquationsPalielināt
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The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.

Recenzijas

This book presents an ambitious overview of modern results and trends in the field of numerical methods for nonlinear PDEs, with an emphasis on the finite element method. The target audience of the book is postgraduates and experienced researchers. this is an excellent monograph describing methods found at the intersection of numerical PDEs and the calculus of variations. (Michael Neilan, SIAM Review, Vol. 58 (3), September, 2016)

This book provides advanced students and experimental researchers with an introduction to numerical methods for nonlinear partial differential equations, in particular those originating from continuum mechanics. This book presents a very nice transition from graduate-level material to state-of-the-art research topics. This is a nice and well-written advanced textbook. (Karsten Urban, Mathematical Reviews, October, 2015)

1 Introduction
1(10)
1.1 Differential Equations and Numerical Methods
1(1)
1.2 Guidelines for the Development of Approximation Schemes
2(1)
1.3 Analytical and Numerical Foundations
3(1)
1.4 Approximation of Classical Formulations
4(3)
1.5 Numerical Methods for Extended Formulations
7(1)
1.6 Objectives and Acknowledgments
8(3)
Part I Analytical and Numerical Foundations
2 Analytical Background
11(34)
2.1 Variational Model Problems
11(7)
2.1.1 Deflection of a Membrane
11(1)
2.1.2 Minimal Surfaces
11(1)
2.1.3 Hyperelastic Materials
12(1)
2.1.4 Obstacle Problems
13(1)
2.1.5 Harmonic Maps
14(1)
2.1.6 Phase-Field Models
14(1)
2.1.7 Plate Bending
15(1)
2.1.8 Crystalline Phase Transitions
15(1)
2.1.9 Free-Discontinuity Problems
16(1)
2.1.10 Segmentation Models
17(1)
2.1.11 Elastoplasticity
17(1)
2.2 Existence of Minimizers
18(11)
2.2.1 The Direct Method in the Calculus of Variations
19(2)
2.2.2 Sobolev Spaces
21(1)
2.2.3 Integral Functionals
22(4)
2.2.4 Existence and Properties of Minimizers
26(3)
2.3 Gradient Flows
29(16)
2.3.1 Differentiation in Banach Spaces
30(1)
2.3.2 Bochner--Sobolev Spaces
31(2)
2.3.3 Existence Theory for Gradient Flows
33(7)
2.3.4 Subdifferential Flows
40(3)
References
43(2)
3 FEM for Linear Problems
45(40)
3.1 Interpolation with Finite Elements
45(10)
3.1.1 Abstract Finite Elements
45(1)
3.1.2 P1 Finite Elements
46(3)
3.1.3 Projection and Quasi-Interpolation Operators
49(3)
3.1.4 Other Estimates
52(3)
3.2 Approximation of the Poisson Problem
55(8)
3.2.1 Variational Formulation
56(1)
3.2.2 Error Estimates
57(3)
3.2.3 Discrete Maximum Principle
60(3)
3.3 Approximation of the Heat Equation
63(14)
3.3.1 Variational Formulation
63(1)
3.3.2 Semidiscrete in Time Approximation
64(4)
3.3.3 Semidiscrete in Space Approximation
68(2)
3.3.4 Fully Discrete Approximation
70(2)
3.3.5 Discrete Maximum Principle
72(3)
3.3.6 A Posteriori Error Estimate
75(2)
3.4 Implementation of the P1 Finite Element Method
77(8)
3.4.1 Poisson Problem
78(2)
3.4.2 Heat Equation
80(4)
References
84(1)
4 Concepts for Discretized Problems
85(42)
4.1 Convergence of Minimizers
85(10)
4.1.1 Failure of Convergence
85(2)
4.1.2 Γ-Convergence of Discretizations
87(1)
4.1.3 Examples of Γ-Convergent Discretizations
88(4)
4.1.4 Error Control for Strongly Convex Problems
92(3)
4.2 Approximation of Equilibrium Points
95(13)
4.2.1 Failure of Convergence
95(1)
4.2.2 Abstract Error Estimates
96(6)
4.2.3 Abstract Subdifferential Flow
102(3)
4.2.4 Weak Continuity Methods
105(3)
4.3 Solution of Discrete Problems
108(19)
4.3.1 Smooth, Unconstrained Minimization
108(4)
4.3.2 Smooth Constrained Minimization
112(4)
4.3.3 Nonsmooth Equations
116(2)
4.3.4 Nonsmooth, Strongly Convex Minimization
118(3)
4.3.5 Nested Iteration
121(2)
References
123(4)
Part II Approximation of Classical Formulations
5 The Obstacle Problem
127(26)
5.1 Analytical Properties
127(10)
5.1.1 Existence and Uniqueness
127(2)
5.1.2 Equivalent Formulations
129(2)
5.1.3 Regularity
131(2)
5.1.4 Penalization
133(2)
5.1.5 Dual Formulation
135(2)
5.2 Finite Element Approximation
137(6)
5.2.1 Abstract Error Analysis
137(2)
5.2.2 Application to Pl-FEM
139(1)
5.2.3 A Posteriori Error Analysis
140(3)
5.3 Iterative Solution Methods
143(10)
5.3.1 Semismooth Newton Iteration
143(3)
5.3.2 Global Primal-Dual Method
146(6)
References
152(1)
6 The Allen-Calm Equation
153(30)
6.1 Analytical Properties
153(14)
6.1.1 Existence and Regularity
154(2)
6.1.2 Stability Estimates
156(7)
6.1.3 Mean Curvature Flow
163(2)
6.1.4 Topological Changes
165(1)
6.1.5 Mass Conservation
166(1)
6.2 Error Analysis
167(7)
6.2.1 Residual Estimate
167(3)
6.2.2 A Priori Error Analysis
170(4)
6.3 Practical Realization
174(9)
6.3.1 Time-Stepping Schemes
175(3)
6.3.2 Computation of the Eigenvalue
178(2)
6.3.3 Implementation
180(2)
References
182(1)
7 Harmonic Maps
183(34)
7.1 Analytical Properties
183(6)
7.1.1 Existence and Nonuniqueness
183(2)
7.1.2 Euler--Lagrange Equations and Nonregularity
185(1)
7.1.3 Compactness
186(2)
7.1.4 Harmonic Map Heat Flow
188(1)
7.2 Numerical Approximation
189(12)
7.2.1 Discrete Harmonic Maps
189(4)
7.2.2 Iterative Computation
193(3)
7.2.3 Projection-Free Iteration
196(3)
7.2.4 Other Target Manifolds
199(1)
7.2.5 Practical Realization
200(1)
7.3 Approximation of Constrained Evolution Problems
201(16)
7.3.1 Harmonic Map Heat Flow
201(2)
7.3.2 Semi-implicit, Linear Schemes
203(6)
7.3.3 Constraint Preservation
209(3)
7.3.4 Approximation of Wave Maps
212(3)
References
215(2)
8 Bending Problems
217(44)
8.1 Mathematical Modeling
217(9)
8.1.1 Bending Models
217(2)
8.1.2 Relations to Hyperelasticity
219(3)
8.1.3 Relations to Linear Elasticity
222(2)
8.1.4 Properties of Isometries
224(2)
8.2 Approximaton of Linear Bending Models
226(8)
8.2.1 Discrete Kirchhoff Triangles
226(4)
8.2.2 Realization
230(2)
8.2.3 Reissner--Mindlin Plate
232(2)
8.3 Approximation of the Nonlinear Kirchhoff Model
234(5)
8.3.1 Discretization
234(2)
8.3.2 Iterative Minimization
236(1)
8.3.3 Realization
237(2)
8.4 Willmore Flow
239(22)
8.4.1 Tangential Differentiation and Curvature
239(5)
8.4.2 Normal Variations
244(5)
8.4.3 Variation of the Willmore Functional
249(1)
8.4.4 Discretization of the Laplace--Beltrami Operator
250(1)
8.4.5 A Numerical Scheme for the Willmore Flow
251(6)
References
257(4)
Part III Methods for Extended Formulations
9 Nonconvexity and Microstructure
261(36)
9.1 Analytical Properties
261(8)
9.1.1 A Scalar Model Problem
262(4)
9.1.2 General Relaxation Result
266(1)
9.1.3 Statistical Description of Oscillations
267(2)
9.2 Direct and Relaxed Numerical Minimization
269(13)
9.2.1 A Lower Bound
269(3)
9.2.2 Upper Bounds
272(3)
9.2.3 Failure of Direct Minimization
275(2)
9.2.4 Approximation of the Relaxed Problem
277(3)
9.2.5 Iterative Minimization
280(2)
9.3 Approximation of Semi-convex Envelopes
282(15)
9.3.1 Upper and Lower Bounds for Wqc
282(3)
9.3.2 Approximation of Wpc
285(3)
9.3.3 Adaptive Computation of Wpc/δr
288(2)
9.3.4 Approximation of Wrc
290(5)
References
295(2)
10 Free Discontinuities
297(36)
10.1 Functions of Bounded Variation
297(10)
10.1.1 Derivatives of Discontinuous Functions
297(3)
10.1.2 Properties of BV(ω)
300(3)
10.1.3 A Variational Model Problem on BV(ω)
303(4)
10.2 Numerical Approximation
307(18)
10.2.1 W1, 1 Conforming Approximation
307(4)
10.2.2 Piecewise Constant Approximation
311(2)
10.2.3 Iterative Solution
313(4)
10.2.4 Realization
317(1)
10.2.5 A Posteriori Error Control
317(3)
10.2.6 Regularized Minimization
320(1)
10.2.7 Total Variation Flow
321(4)
10.3 Segmentation
325(8)
10.3.1 The Mumford--Shah Functional
325(2)
10.3.2 Regularization of I'(μ)
327(1)
10.3.3 Numerical Approximation of ATε
328(1)
10.3.4 The Perona--Malik Equation
329(3)
References
332(1)
11 Elastoplasticity
333(32)
11.1 Modeling and Analytical Properties
333(9)
11.1.1 One-Dimensional Plastic Effects
333(1)
11.1.2 Hypotheses of Multi-dimensional Elastoplasticity
334(2)
11.1.3 Mathematical Model
336(1)
11.1.4 Flow Rules and Coercivity
337(3)
11.1.5 Equivalent Formulations and Existence
340(2)
11.2 Approximation of Rate-Independent Evolutions
342(10)
11.2.1 Time-Incremental Minimization
343(3)
11.2.2 Discretization in Space
346(2)
11.2.3 Fully Discrete Approximation
348(2)
11.2.4 A Posteriori Error Control
350(2)
11.3 Numerical Solution
352(13)
11.3.1 Solution of the Discretized Flow Rule
352(4)
11.3.2 Newton Method for Nonlinear Elasticity
356(2)
11.3.3 Implementation of Elastoplasticity
358(5)
References
363(2)
Appendix A Auxiliary Routines 365(20)
Appendix B Frequently Used Notation 385(6)
Index 391
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