Table Of ContentSeries on Advances in Mathematics for Applied Sciences - Vol. 59
NUMERICAL METHODS
FOR VISCOSITY SOLUTIONS
AND APPLICATIONS
Editors
Maurizio Falcone
Charalampos Makridakis
World Scientific
NUMERICAL METHODS
FOR VISCOSITY SOLUTIONS
AND APPLICATIONS
Series on Advances in Mathematics for Applied Sciences - Vol. 59
NUMERICAL METHODS
FOR VISCOSITY SOLUTIONS
AND APPLICATIONS
Editors
Maurizio Falcone
Universita di Roma "La Sapienza"
Charalampos Makridakis
University of Crete
m World Scientific
ll Singapore * New Jersey • London • Hong Kong
Published by
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NUMERICAL METHODS FOR VISCOSITY SOLUTIONS AND APPLICATIONS
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.
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V
FOREWORD
The notion of viscosity solution, introduced by M. Crandall and P. L. Lions
more than ten years ago, has given a new tool for the analysis of partial diffe
rential equations. This notion of weak solution allows to treat f.e. hyperbolic
equations with nin differentiable or discontinuous coefficients. Moreover, by
the so-called "level set" approach is possible to track the evolution of interfaces
after the on-set of singularities just looking for viscosity solutions of Hamilton-
Jacobi equation. The above features have shown to be very effective in many
fields of application including optimal control, image processing, geometrical
optics and front propagation.
Naturally, to deal with those applications it is necessary to develop efficient
and fast algorithms. Several methods have been proposed, starting from first
order problems where finite differences schemes for Hamilton-Jacobi equations
can be obtained simply integrating in space the schemes for conservation laws.
However, this simple solution is valid only for one dimensional problems so
further analysis is needed to deal with real applications. The numerical analysis
of Hamilton-Jacobi equations has progressed rapidly in the last ten years and
now approximation schemes based on finite elements and finite volumes are also
available. Moreover, the complexity of the algorithms has increased trying to
get high-order accuracy in the approximation of the solutions and to include
advanced computational tools as adaptive grid refinements. This is an active
reasearch field with challenging applications.
We hope this volume can attract readers involved in the numerical appro
ximation of differential models in the above mentioned fields of applications,
engineers, graduate students as well as researchers in numerical analysis. It
contains twelve papers dealing with the approximation of first and second order
problems for a variety of applications. Some of them present new algorithms
and deal with technical issues related for their implementation. Many test
problems have been examined to evaluate the performances of the algorithms.
Other contributions are more theoretical, dealing with the convergence of ap
proximation schemes. Some of the papers were presented at a Workshop held
in 1999 at the Institute of Applied and Computational Mathematics, FORTH
in Heraklion under the auspices of the EC TMR Project "Viscosity solutions
vi
and Applications". We take this opportunity to thank the people at FORTH
for their warm hospitality and support.
May 15, 2001
M. Falcone
Dipartimento di Matematica
Universita di Roma "La Sapienza"
P. Aldo Moro 2, 00185 - Roma, Italy
e-mail: [email protected]
Ch. Makridakis
Department of Applied Mathematics
University of Crete, 71409 Heraklion-Crete, Greece
and Institute of Applied and Computational Mathematics
FORTH, 71110 Heraklion - Crete, Greece
e-mail: [email protected]
VII
CONTENTS
Foreword v
Geometrical Optics and Viscosity Solutions 1
A.-P. Blanc, G. T. Kossioris and G. N. Makrakis
Computation of Vorticity Evolution for a Cylindrical 21
Type-II Superconductor Subject to Parallel and
Transverse Applied Magnetic Fields
A. Briggs, J. Claisse, C. M. Elliott and V. Styles
A Characterization of the Value Function for a Class 47
of Degenerate Control Problems
F. Camilli
Some Microstructures in Three Dimensions 59
M. Chipot and V. Lecuyer
Convergence of Numerical Schemes for the Approximation 77
of Level Set Solutions to Mean Curvature Flow
K. Deckelnick and G. Dziuk
Optimal Discretization Steps in Semi-Lagrangian Approximation 95
of First-Order PDEs
M. Falcone, R. Ferretti and T. Manfroni
Convergence Past Singularities to the Forced Mean 119
Curvature Flow for a Modified Reaction-Diffusion Approach
F. Fierro
The Viscosity-Duality Solutions Approach to Geometric Optics 133
for the Helmholtz Equation
L. Gosse and F. James
Adaptive Grid Generation for Evolutive 153
Hamilton-Jacobi-Bellman Equations
L. Grime
VIII
Solution and Application of Anisotropic Curvature Driven
Evolution of Curves (and Surfaces)
K. Mikula
An Adaptive Scheme on Unstructured Grids for the
Shape-From-Shading Problem
M. Sagona and A. Seghini
On a Posteriori Error Estimation for
Constant Obstacle Problems
A. Veeser
1
GEOMETRICAL OPTICS AND VISCOSITY SOLUTIONS
A.-P. BLANC
Universitat de Illes Balears, Spain
E-mail:ablanc@serdis. dis.ulpgc. es
G.T. KOSSIORIS
Dept Mathematics, Univ. Crete, Greece &
Institute of Applied and Computational Mathematics,
FORTH, P.O. Box 1527, Heraklion 711 10, Greece
E-mail:kosioris ©zargana. math. uch. gr
G.N. MAKRAKIS
Institute of Applied and Computational Mathematics,
FORTH, P.O. Box 1527, Heraklion 711 10, Greece
E-mail:makrakg Qiacm.forth. gr
In the present paper we use the method of relevant functions to show a way for
computing the correct asymptotic solution of the Helmholtz equation near caus
tics, provided that the various branches of the multivalued solution of the eikonal
equation are given. We also describe how to compute the multivalued solution to
the eikonal equation, by using viscosity solution eikonal solvers. Finally, we make
some remarks on the role played by the viscosity solution in shadow zone.
1 High frequency waves and WKB method
We consider the propagation of two-dimensional time-harmonic scalar waves
in a medium with variable refraction index n(x) = co/c(x), CQ being the
reference wave velocity and c(x) the velocity at the point x = (x, z) £ D, D
being some unbounded domain of EQ. We assume that n 6 C°°(1R^) and
n > 0. The two-dimensional wave field u(x, k) is governed by the Helmholtz
equation
Au + fc2n2(x)u(x,fc) = /(x,fc), X G D, (1.1)
where k = UI/CQ is the wavenumber (ui is the frequency of the waves) and
/(x, A;) represents a compactly supported source generating the waves. We
are interested in the asymptotic behavior of u(x, k) as k —> oo, (i.e. for very
large frequencies a;), assuming that x remains in a compact subset of D and
outside the support of the source function /.
Note that the asymptotic decomposition of scattering solutions when si
multaneously |x| and k go to infinity is a rather complicated problem, as, in
