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Variational Problems, the Ritz Method, and the Idea of Orthogonality

  • Eberhard Zeidler

Keywords

Variational Problem Monotone Operator Cauchy Sequence Ritz Method Elliptic Differential Equation 
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References to the Literature

  1. Classical works: Euler (1744) and Lagrange (1762) (foundation of the calculus of variations), Laplace (1782) (Laplace equation), Gauss (1839) (potential theory), Riemann (1851), (1857) (foundation of complex function theory), Weierstrass (1870) (criticism of the Dirichlet principle), Hilbert (1901) (justification of the Dirichlet principle), Friedrichs (1934), Sobolev (1936), Courant and Hilbert (1937, M), Schwartz (1950) (distributions).Google Scholar
  2. Collected works: Euler (1911), Vols. 1-72, Lagrange (1867), Vols. 1-14, Laplace (1878), Vols. 1-14, Gauss (1863), Vols. 1-12, Riemann (1892), Vols. 1-2, Weierstrass (1894), Vols. 1-7, Poincaré (1916), Vols. 1-11, Hilbert (1932), Vols. 1-3.Google Scholar
  3. History of mathematics of the nineteenth century: Klein (1926, M).Google Scholar
  4. History of functional analysis: Dieudonné (1981, M).Google Scholar
  5. History of calculus from Euler to Riemann and Weierstrass: Bottazzini (1986, M).Google Scholar
  6. Hilbert’s problems, their solution, and their influence on the mathematics of the twentieth century: Aleksandrov (1971, S), Browder (1976, S).Google Scholar
  7. Biographies of Hilbert: Blumenthal (1932), Reid (1970).Google Scholar
  8. History of potential theory and of the theory of partial differential equations: Burkhardt and Meyer (1900, S), Sommerfeld (1900, S), and Lichtenstein (1921, S) (three articles in the encyclopedia of mathematics), Dieudonné (1981, M).Google Scholar
  9. Collection of important classical papers in analysis: Birkhoff (1973).Google Scholar
  10. Potential theory, integral equations, and boundary value problems: Courant and Hilbert (1953, M), Günter (1957, M).Google Scholar
  11. Classical mathematical physics: Courant and Hilbert (1953, M), Vols. 1, 2.Google Scholar
  12. Modern mathematical physics: Reed and Simon (1971, M), Vols. 1–4.Google Scholar
  13. Interpolation theory and linear partial differential equations: Lions and Magenes (1968, M), Vols. 1–3, Triebel (1978, M), (1983, M).Google Scholar
  14. General theory of linear differential operators, pseudodifferential operators, and Fourier integral operators: Hörmander (1983, M), Vols. 1–4.Google Scholar
  15. Variational methods: Courant and Hilbert (1953, M) (recommended as an introduction), Morrey (1966, M) (standard work), Klötzler (1971, M), Veite (1976, M) (elementary introduction), Blanchard and Brüning (1982, M), Giaquinta and Hildebrandt (1989, M) (modem standard work).Google Scholar
  16. Survey on minimal surfaces: Osserman (1986, M).Google Scholar
  17. Minimal surfaces, modem geometric measure theory, and functions of bounded variations: Giusti (1984, M).Google Scholar
  18. Harmonic mappings: Jost (1984, L), (1985, L), (1988, L), Hildebrandt (1985, L), Hildebrandt and Leis (1989, S).Google Scholar
  19. Essays on mathematics and optimal form: Hildebrandt and Tromba (1985, M).Google Scholar
  20. Regularity theory: Necas (1982, S), (1983, M) and Giaquinta (1983, M) (introduction), Ladyženskaja and Uralceva (1964, M), (1973, M), Morrey (1966, M), Friedman (1969, M) (linear equations), Gilbarg and Trudinger (1983, M), Koshelev (1985, L), Giaquinta and Hildebrandt (1989, M) (cf. also Problem 18.7).Google Scholar
  21. Classical papers on the Ritz method: Ritz (1909), Courant (1943).Google Scholar
  22. Numerical realization of the Ritz method: Veite (1976, M), Stoer and Bulirsch (1976, M) (elementary introduction), Michlin (1962, M), (1969, M), (1985, M), Rektorys (1977, M), Glowinski (1984, M), Hackbusch (1986, M).Google Scholar
  23. The Ritz method and the method of finite elements: Ciarlet (1977, M). Cf. also the References to the Literature to the Appendix.Google Scholar
  24. Multigrid methods: Hackbusch (1985, M).Google Scholar
  25. Software system ELLPACK for solving elliptic equations on computers: Rice and Boisvert (1984, M) (Ritz method and finite elements, finite differences, SFT, multigrid methods, ect.).Google Scholar
  26. Software system for general classes of mathematical problems: IMSL (1987).Google Scholar
  27. Handbook of numerical analysis: Ciarlet and Lions (1988, M), Vols. 1 ff (finite element method, difference method, etc.).Google Scholar
  28. Supercomputing: Murman (1985, P), Lichnewsky and Saguez (1987, S), Martin (1988, S).Google Scholar
  29. Numerical recipies—the art of scientific computing: Press (1986, M).Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Eberhard Zeidler
    • 1
  1. 1.Max-Planck-Institut fuer Mathematik in den NaturwissenschaftenLeipzigGermany

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