Finite element interpolation error bounds with applications to eigenvalue problems

  • Peter Arbenz
Original Papers


Optimal finite element interpolation eror bounds are presented for piecewise linear, quadratic and Hermite-cubic elements in one dimenson. These bounds can be used to compute upper and lower bounds for eigenvalues of second and fourth order elliptic problems. Numerical computations demonstrate the usefulness of the theoretical results.


Lower Bound Numerical Computation Theoretical Result Mathematical Method Eigenvalue Problem 
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Es werden optimale Fehlerschranken für die eindimensionale finite Element-Interpolation mit stückweise linearen, quadratischen und Hermite-kubischen Elementen angegeben. Diese Schranken können dazu verwendet werden, unter und obere Schranken für Eigenwerte von elliptischen Problemen 2. und 4. Ordnung zu berechnen. Dazu werden numerische Resultate angeführt, welche die Nützlichkeit der theoretischen Resultate zeigen.


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  1. [1]
    P. Arbenz,Computable finite element error bounds for Poisson's equation. IMA J. Numer. Anal. (to appear).Google Scholar
  2. [2]
    R. E. Barnhill, J. H. Brown and A. R. Mitchell,A comparison of finite element error bounds for Poisson's equation. IMA J. Numer. Anal.1, 95–103 (1981).Google Scholar
  3. [3]
    N. W. Bazley and D. W. Fox,Methods for lower bounds to frequencies of continuous elastic systems. Z. Angew. Math. Phys.17, 1–37 (1966).Google Scholar
  4. [4]
    G. Birkhoff, C. de Boor, B. Swarts and B. Wendroff,Rayleigh-Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal.3, 188–203 (1966).Google Scholar
  5. [5]
    G. Birkhoff and G. C. Rota,Ordinary differential equations. (3. ed.) Wiley, New York 1978.Google Scholar
  6. [6]
    G. Birkhoff, M. H. Schultz and R. S. Varga,Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math.11, 232–256 (1968).Google Scholar
  7. [7]
    P. G. Ciarlet,The finite element method for elliptic problems. North-Holland, Amsterdam-New York-Oxford 1978.Google Scholar
  8. [8]
    P. G. Ciarlet, M. H. Schultz and R. S. Varga,Numerical methods of high-order accuracy for nonlinear boundary value problems. III Eigenvalue problems. Numer Math.12, 120–133 (1968).Google Scholar
  9. [9]
    J. Dieudonné,Eléments d'analyse I. Gauthiers-Villars, Paris 1969.Google Scholar
  10. [10]
    J. G. Pierce and R. S. Varga,Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. I: Estimates relating Rayleigh-Ritz and Galerkin approximations to eigenfunctions. SIAM J. Numer. Anal.9, 137–151 (1972).Google Scholar
  11. [11]
    S. W. Schoombie and J. F. Botha,Error estimates for the solution of the radial Schrödinger equation by the Rayleigh-Ritz finite element method. IMA J. Numer. Anal.1, 47–63 (1981).Google Scholar
  12. [12]
    M. H. Schultz,Error bounds for polynomial spline interpolation. Math. Comp.24, 507–515 (1970).Google Scholar
  13. [13]
    M. H. Schultz,L 2-error bounds for the Rayleigh-Ritz-Galerkin method. SIAM J. Numer. Anal.8, 737–748 (1971).Google Scholar
  14. [14]
    H. R. Schwarz,Methode der finiten Elemente. Teubner, Stuttgart 1980.Google Scholar
  15. [15]
    V. G. Sigillito,Explicit a priori inequalities with applications to boundary value problems. Research notes in mathematics 13. Pitman, London-San Francisco-Melbourne 1977.Google Scholar
  16. [16]
    W. Walter,Gewöhnliche Differentialgleichungen. Springer, Berlin-Heidelberg-New York 1972.Google Scholar
  17. [17]
    H. F. Weinberger,Variational methods for eigenvalue problems. Regional conference series in applied mathematics 15. SIAM, Philadelphia 1974.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1983

Authors and Affiliations

  • Peter Arbenz
    • 1
  1. 1.Institut für angewandte MathematikUniversität ZürichZürich

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