Theory and Numerics of Differential Equations

by Blowey, James; Coleman, John Patrick; Craig, Alan W.; Craig, Alan W.

• ISBN13: 9783540418467
• ISBN10: 3540418466
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: Springer Verlag

Summary

This book contains detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians who require a succinct and accurate account of recent research in areas parallel to their own, and graduates in mathematical sciences.

Table of Contents

Preface	v
Contents	vii
Spectral, Spectral Element and Mortar Element Methods	1	(58)
Christine Bernardi
Yvon Maday
Introduction	1	(2)
Spectral Methods in Tensorized Geometries	3	(33)
Legendre Polynomials, and Polynomial Approximation	3	(5)
Gauss-Lobatto Formula, and Polynomial Interpolation	8	(4)
Spectral Discretization of the Laplace Equation	12	(5)
Spectral Discretization of the Stokes and Navier-Stokes Equations	17	(9)
Polynomial Approximation of Discontinuous Functions	26	(5)
Spectral Discretization of Hyperbolic Equations	31	(5)
Extension to more Complex Geometries	36	(23)
The Mortar Spectral Element Method	36	(11)
Spectral Discretization in a Cylinder	47	(6)
References	53	(6)
Numerical Analysis of Microstructure	59	(68)
Carsten Carstensen
Motivation and Overview	60	(4)
Four Minimization Problems in Rn	64	(2)
Function Spaces	66	(1)
Four Minimization Problems in W1, p(0, 1)	67	(3)
The Direct Method in the Calculus of Variations	70	(5)
First Stage: Construction of Infimizing Sequences	71	(1)
Second Stage: a Priori Bounds of Infimizing Sequences	72	(1)
Third Stage: Passage to the Limit	73	(2)
Typical Information from Infimizing Sequences	75	(2)
Finite Element Discretization of (M)	77	(8)
Energy Minimization Rates	85	(1)
Cluster of Local Minimizers	86	(4)
Results in Higher Dimensions and on Real Applications	90	(4)
Scalar 2-well Problem with Linear Growth	90	(1)
Example in Optimal Design	90	(1)
4-well Problem due to Tartar	91	(1)
Examples from Crystal Physics	91	(1)
Example from Micromagnetics	92	(2)
Young Measure Approximations	94	(4)
Weak Convergence and Generated Young Measure	98	(2)
Young Measure Relaxation	100	(5)
Convexification	105	(2)
Equivalence	107	(3)
Error Estimates for (Ch)	110	(6)
Numerical Algorithms	116	(2)
Model of Phase Transitions	118	(4)
Summary	122	(5)
References	124	(3)
Maple for Stochastic Differential Equations	127	(52)
S. Cyganowski
L. Grune
P.E. Kloeden
Introduction	127	(1)
Stochastic Differential Equations	128	(2)
Terminology	129	(1)
The MAPLE Software Package ``stochastic''	130	(2)
MAPLE-terminology	131	(1)
Ito Stchastic Calculus	132	(5)
Partial Differential Operators	132	(1)
LO Operator routine	132	(1)
LJ Operator routine	133	(1)
Combined MLJ Operator routine	134	(1)
Ito Formula	135	(1)
LFP Operator: Fokker-Planck Equation	136	(1)
Stratonovich Stochastic Calculus	137	(5)
Ito-Stratonovich Drift Correction Procedures	138	(2)
Stratonovich L0 Operator	140	(1)
Staratonovich Chain Rule Transformation	141	(1)
Explicitly Solvable Scalar SDEs	142	(5)
Linearsde Routine	142	(2)
Reducible Routine	144	(2)
Explicit Routine	146	(1)
Linear Vector SDEs	147	(10)
Linearization	148	(1)
Spherical Coordinates	149	(3)
Second Moment Equation	152	(1)
The procedures ``pmatrix2pvector'' and ``pvector2pmatrix''	153	(2)
Subprocedures for ``momenteqn''	155	(2)
Commutative and Coloured Noise	157	(4)
Communtative Noise of 1st Kind	157	(1)
Commutative Noise of 2nd Kind	158	(2)
Coloured Noise	160	(1)
Strong Numerical Schemes	161	(9)
Euler Scheme	161	(2)
Milstein Scheme	163	(1)
Milstein Scheme for Commutative Noise	164	(2)
Order 1.5 Strong Stochastic Taylor Scheme	166	(2)
Order 2.0 Strong Stochastic Taylor Scheme	168	(2)
Weak Numerical Schemes	170	(9)
Weak Euler Scheme	170	(2)
Order 2.0 Weak Stochastic Taylar Scheme	172	(2)
Order 3.0 Weak Stochastic Taylor Scheme	174	(2)
References	176	(3)
Nonlinear Multigrid Techniques	179	(52)
Ralf Kornhuber
Introduction	179	(2)
Self-Adjoint Linear Problems	181	(9)
Continuous Problem and Discretiztion	181	(2)
Successive Subspace Correction and Multigrid	183	(5)
Concluding Remarks	188	(2)
Smooth Nonlinear Problems	190	(18)
Continuous Problem and Discretization	190	(4)
Inexact Newton Methods	194	(6)
Newton Multigrid with Nested Iteration	200	(4)
Nonlinear Multigrid and FAS	204	(2)
Concluding Remarks	206	(2)
Piecewise Smooth Semilinear Problems	208	(23)
Continuous Problem and Discretization	208	(3)
Obstacle Problems	211	(7)
Constrained Newton Linearization and Monotone Multigrid	218	(6)
Concluding Remarks	224	(1)
References	225	(6)
Hyperbolic Differential Equations and Adaptive Numerics	231	(47)
Kyoung-Sook Moon
Anders Szepessy
Raul Tempone
Georgios Zouraris
Hyperbolic Conservation Laws	231	(16)
Introduction	231	(3)
Convergence with Measure Valued Solutions	234	(6)
Convergence of a Finite Volume Method	240	(2)
A Uniform Bound in L2	242	(3)
Entropy Consistency	245	(1)
Consistency with Initial Data	246	(1)
Adaptive Numerics for Differential Equations	247	(31)
Global and Local Errors for Differential Equations	247	(1)
A Variation Principle for Errors in ODE	248	(4)
Adaptive Algorithms	252	(4)
Numerical Experiments for ODE	256	(2)
A Variational Principle for Errors in PDE	258	(6)
An a Posteriori Expansion of the Global Error	264	(3)
Weak Approximation of SDE	267	(7)
Numerical Experiments for SDE	274	(4)
References	278