From finite differences to finite elements. A short history of numerical analysis of partial differential equations (V. Thomée). Orthogonal spline collocation methods for partial differential equations (B. Bialecki, G. Fairweather). Spectral methods for hyperbolic problems (D. Gottlieb, J.S. Hesthaven). Wavelet methods for PDEs N some recent developments (W. Dahmen). Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws (B. Cockburn). Adaptive Galerkin finite element methods for partial differential equations (R. Rannacher). The p and hp finite element method for problems on thin domains (M. Suri). Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow (D. Silvester, H. Elman, D. Kay, A. Wathen). A review of algebraic multigrid (K. Stüben). Geometric multigrid with applications to computational fluid dynamics (P. Wesseling, C.W. Oosterlee). The method of subspace corrections (J. Xu). Moving finite element, least squares, and finite volume approximations of steady and time-dependent PDEs in multidimensions (M.J. Baines). Adaptive mesh movement - the MMPDE approach and its applications (W. Huang, R.D. Russell). The geometric integration of scale-invariant ordinary and partial differential equations (C.J. Budd, M.D. Piggott). A summary of numerical methods for time-dependent advection-dominated partial differential equations (R.E. Ewing, H. Wang). Approximate factorization for time-dependent partial differential equations (P.J. van der Houwen, B.P. Sommeijer).
/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 Thomée 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. Refe
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- © North Holland 2001
- 10th July 2001
- North Holland
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University of Strathclyde, Glasgow, G1 1XH, Scotland, UK
Katholieke Universiteit Leuven, Leuven (Haverlee), B-3001, Belgium
Oxford University, Oxford, OX1 3QD, UK