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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Partial Differential Equations, Volume 7 (Paperback)
Loot Price: R3,125
Discovery Miles 31 250
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Partial Differential Equations, Volume 7 (Paperback)
Series: Numerical Analysis 2000
Expected to ship within 10 - 15 working days
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/homepage/sac/cam/na2000/index.html7-Volume Set now available at
special set price ! Over the second half of the 20th century the
subject area loosely referred to as numerical analysis of partial
differential equations (PDEs) has undergone unprecedented
development. At its practical end, the vigorous growth and steady
diversification of the field were stimulated by the demand for
accurate and reliable tools for computational modelling in physical
sciences and engineering, and by the rapid development of computer
hardware and architecture. At the more theoretical end, the
analytical insight into the underlying stability and accuracy
properties of computational algorithms for PDEs was deepened by
building upon recent progress in mathematical analysis and in the
theory of PDEs. To embark on a comprehensive review of the field of
numerical analysis of partial differential equations within a
single volume of this journal would have been an impossible task.
Indeed, the 16 contributions included here, by some of the foremost
world authorities in the subject, represent only a small sample of
the major developments. We hope that these articles will,
nevertheless, provide the reader with a stimulating glimpse into
this diverse, exciting and important field. The opening paper by
Thomee reviews the history of numerical analysis of PDEs, starting
with the 1928 paper by Courant, Friedrichs and Lewy on the solution
of problems of mathematical physics by means of finite differences.
This excellent survey takes the reader through the development of
finite differences for elliptic problems from the 1930s, and the
intense study of finite differences for general initial value
problems during the 1950s and 1960s. The formulation of the concept
of stability is explored in the Lax equivalence theorem and the
Kreiss matrix lemmas. Reference is made to the introduction of the
finite element method by structural engineers, and a description is
given of the subsequent development and mathematical analysis of
the finite element method with piecewise polynomial approximating
functions. The penultimate section of Thomee's survey deals with
`other classes of approximation methods', and this covers methods
such as collocation methods, spectral methods, finite volume
methods and boundary integral methods. The final section is devoted
to numerical linear algebra for elliptic problems. The next three
papers, by Bialecki and Fairweather, Hesthaven and Gottlieb and
Dahmen, describe, respectively, spline collocation methods,
spectral methods and wavelet methods. The work by Bialecki and
Fairweather is a comprehensive overview of orthogonal spline
collocation from its first appearance to the latest mathematical
developments and applications. The emphasis throughout is on
problems in two space dimensions. The paper by Hesthaven and
Gottlieb presents a review of Fourier and Chebyshev pseudospectral
methods for the solution of hyperbolic PDEs. Particular emphasis is
placed on the treatment of boundaries, stability of time
discretisations, treatment of non-smooth solutions and multidomain
techniques. The paper gives a clear view of the advances that have
been made over the last decade in solving hyperbolic problems by
means of spectral methods, but it shows that many critical issues
remain open. The paper by Dahmen reviews the recent rapid growth in
the use of wavelet methods for PDEs. The author focuses on the use
of adaptivity, where significant successes have recently been
achieved. He describes the potential weaknesses of wavelet methods
as well as the perceived strengths, thus giving a balanced view
that should encourage the study of wavelet methods. Aspects of
finite element methods and adaptivity are dealt with in the three
papers by Cockburn, Rannacher and Suri. The paper by Cockburn is
concerned with the development and analysis of discontinuous
Galerkin (DG) finite element methods for hyperbolic problems. It
reviews the key properties of DG methods for nonlinear hyperbolic
conservation laws from a novel viewpoint that stems from the
observation that hyperbolic conservation laws are normally arrived
at via model reduction, by elimination of dissipation terms.
Rannacher's paper is a first-rate survey of duality-based a
posteriori error estimation and mesh adaptivity for Galerkin finite
element approximations of PDEs. The approach is illustrated for
simple examples of linear and nonlinear PDEs, including also an
optimal control problem. Several open questions are identified such
as the efficient determination of the dual solution, especially in
the presence of oscillatory solutions. The paper by Suri is a lucid
overview of the relative merits of the hp and p versions of the
finite element method over the h version. The work is presented in
a non-technical manner by focusing on a class of problems concerned
with linear elasticity posed on thin domains. This type of problem
is of considerable practical interest and it generates a number of
significant theoretical problems. Iterative methods and multigrid
techniques are reviewed in a paper by Silvester, Elman, Kay and
Wathen, and in three papers by Stuben, Wesseling and Oosterlee and
Xu. The paper by Silvester et al. outlines a new class of robust
and efficient methods for solving linear algebraic systems that
arise in the linearisation and operator splitting of the
Navier-Stokes equations. A general preconditioning strategy is
described that uses a multigrid V-cycle for the scalar
convection-diffusion operator and a multigrid V-cycle for a
pressure Poisson operator. This two-stage approach gives rise to a
solver that is robust with respect to time-step-variation and for
which the convergence rate is independent of the grid. The paper by
Stuben gives a detailed overview of algebraic multigrid. This is a
hierarchical and matrix-based approach to the solution of large,
sparse, unstructured linear systems of equations. It may be applied
to yield efficient solvers for elliptic PDEs discretised on
unstructured grids. The author shows why this is likely to be an
active and exciting area of research for several years in the new
millennium. The paper by Wesseling and Oosterlee reviews geometric
multigrid methods, with emphasis on applications in computational
fluid dynamics (CFD). The paper is not an introduction to
multigrid: it is more appropriately described as a refresher paper
for practitioners who have some basic knowledge of multigrid
methods and CFD. The authors point out that textbook multigrid
efficiency cannot yet be achieved for all CFD problems and that the
demands of engineering applications are focusing research in
interesting new directions. Semi-coarsening, adaptivity and
generalisation to unstructured grids are becoming more important.
The paper by Xu presents an overview of methods for solving linear
algebraic systems based on subspace corrections. The method is
motivated by a discussion of the local behaviour of high-frequency
components in the solution of an elliptic problem. Of novel
interest is the demonstration that the method of subspace
corrections is closely related to von Neumann's method of
alternating projections. This raises the question as to whether
certain error estimates for alternating directions that are
available in the literature may be used to derive convergence
estimates for multigrid and/or domain decomposition methods. Moving
finite element methods and moving mesh methods are presented,
respectively, in the papers by Baines and Huang and Russell. The
paper by Baines reviews recent advances in Galerkin and
least-squares methods for solving first- and second-order PDEs with
moving nodes in multidimensions. The methods use unstructured
meshes and they minimise the norm of the residual of the PDE over
both the computed solution and the nodal positions. The
relationship between the moving finite element method and L2
least-squares methods is discussed. The paper also describes moving
finite volume and discrete l2 least-squares methods. Huang and
Russell review a class of moving mesh algorithms based upon a
moving mesh partial differential equation (MMPDE). The authors are
leading players in this research area, and the paper is largely a
review of their own work in developing viable MMPDEs and efficient
solution strategies. The remaining three papers in this special
issue are by Budd and Piggott, Ewing and Wang and van der Houwen
and Sommeijer. The paper by Budd and Piggott on geometric
integration is a survey of adaptive methods and scaling invariance
for discretisations of ordinary and partial differential equations.
The authors have succeeded in presenting a readable account of
material that combines abstract concepts and practical scientific
computing. Geometric integration is a new and rapidly growing area
which deals with the derivation of numerical methods for
differential equations that incorporate qualitative information in
their structure. Qualitative features that may be present in PDEs
might include symmetries, asymptotics, invariants or orderings and
the objective is to take these properties into account in deriving
discretisations. The paper by Ewing and Wang gives a brief summary
of numerical methods for advection-dominated PDEs. Models arising
in porous medium fluid flow are presented to motivate the study of
the advection-dominated flows. The numerical methods reviewed are
applicable not only to porous medium flow problems but second-order
PDEs with dominant hyperbolic behaviour in general. The paper by
van der Houwen and Sommeijer deals with approximate factorisation
for time-dependent PDEs. The paper begins with some historical
notes and it proceeds to present various approximate factorisation
techniques. The objective is to show that the linear system arising
from linearisation and discretisation of the PDE may be solved more
efficiently if the coefficient matrix is replaced by an approximate
factorisation based on splitting. The paper presents a number of
new stability results obtained by the group at CWI Amsterdam for
the resulting time integration methods.
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