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Stability for Discretisations of Elliptic Problems

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The Concept of Stability in Numerical Mathematics

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 45))

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Abstract

In the previous chapter we treated partial differential equations of hyperbolic and parabolic type. The third type of elliptic differential equations is considered in this chapter.

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References

  1. Aubin, J.P.: Behaviour of the error of the approximate solution of boundary value problems for linear operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21, 599–637 (1967)

    MATH  MathSciNet  Google Scholar 

  2. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics,3rd ed. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  3. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Discontinuous Galerkin Methods. Theory, Computation and Applications, Lect. Notes Comput. Sci. Eng., Vol. 11. Springer, Berlin (2000).

    Google Scholar 

  4. Fréchet, M.: Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci. Paris 144, 1414–1416 (1907)

    MATH  Google Scholar 

  5. Hackbusch, W.: Bemerkungen zur iterativen Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren. Rev. Roumaine Math. Pures Appl. 26, 1319–1329 (1981)

    MATH  MathSciNet  Google Scholar 

  6. Hackbusch, W.: On the regularity of difference schemes. Ark. Mat. 19, 71–95 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New York (1994)

    Book  MATH  Google Scholar 

  8. Hackbusch, W.: Elliptic differential equations. Theory and numerical treatment, Springer Series in Computational Mathematics, Vol. 18, 2nd ed. Springer, Berlin (2003)

    Google Scholar 

  9. Hackbusch, W.: Multi-grid methods and applications, Springer Series in Computational Mathematics, Vol. 4. Springer, Berlin (2003)

    Google Scholar 

  10. Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen, 3rd ed. (2005). www.mis.mpg.de/scicomp/Fulltext/EllDgl.ps

    Google Scholar 

  11. Hemker, P.W.: Mixed defect correction iteration for the solution of a singular perturbation problem. Comput. Suppl. 5, 123–145 (1984)

    Article  MathSciNet  Google Scholar 

  12. John, V., Maubach, J.M., Tobiska, L.: Nonconforming streamline-diffusion-finite-element-methods for convection diffusion problems. Numer. Math. 78, 165–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jovanovič, B., Süli, E.: Analysis of finite difference schemes for linear partial differential equations with generalized solutions, Springer Series in Computational Mathematics, Vol. 45. Springer, London (2013)

    Google Scholar 

  14. Kanschat, G.: Discontinuous Galerkin methods for viscous incompressible flows. Advances in Numerical Mathematics. Deutscher Universitätsverlag, Wiesbaden (2007)

    Google Scholar 

  15. Nečas, J.: Sur la coercivité des formes sesquilinéares elliptiques. Rev. Roumaine Math. Pures Appl. 9, 47–69 (1964)

    MATH  MathSciNet  Google Scholar 

  16. Nitsche, J.: Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer. Math. 11, 346–348 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ostrowski, A.M.: Über die Determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv. 10, 69–96 (1937)

    Article  MathSciNet  Google Scholar 

  18. Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods. Springer, Berlin (2011)

    Google Scholar 

  19. Riesz, F.: Sur une espèce de géométrie analytique des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411 (1907)

    MATH  Google Scholar 

  20. Riesz, F., Sz.-Nagy, B.: Functional Analysis. Dover Publ. Inc, New York (1990)

    Google Scholar 

  21. Roos, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations: Convection-diffusion and flow problems, Springer Series in Computational Mathematics, Vol. 24. Springer, Berlin (1996)

    Book  Google Scholar 

  22. Shi, Z.C.: A convergence condition for the quadrilateral Wilson element. Numer. Math. 44, 349–361 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. Stummel, F.: The generalized patch test. SIAM J. Numer. Anal. 16, 449–471 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stummel, F.: The limitations of the patch test. Int. J. Num. Meth. Engng. 15, 177–188 (1980)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. MPI für Mathematik in den Naturwissenschaften, Leipzig, Germany

    Wolfgang Hackbusch

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  1. Wolfgang Hackbusch
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Hackbusch, W. (2014). Stability for Discretisations of Elliptic Problems. In: The Concept of Stability in Numerical Mathematics. Springer Series in Computational Mathematics, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39386-0_7

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