Approximation of the eigenvalue problem for elliptic operator by finite element method with numerical integration

• Autorzy: Lewińska Ewa

Identyfikatory

DOI

10.14708/ma.v23i37.1824 Abstract

• Języki publikacji: EN

Abstrakty

EN

The author considers the effect of numerical integration in the case of solving a two-dimensional eigenvalue problem for the second-order elliptic differential operator via the finite element method. It is proved that the optimal estimates for eigenfunctions (namely, the estimates of the same order as the optimal estimates for the classical finite element approximation without numerical integration) are valid under the assumption that the precision of the numerical quadrature is the same as that for the corresponding boundary value problem.

Słowa kluczowe

EN

eigenvalue problems

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Matematyka Stosowana : matematyka dla społeczeństwa
• Rocznik: 1994
• Tom: Nr 37
• Strony: 3--14
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Lewińska Ewa

• Instytut Matematyki, Politechnika Warszawska, Plac Politechniki 1, 00-661 Warszawa

Bibliografia

• [1] U. Banerjee, A Note on the Effect of Numerical Quadrature in Finite Element Eigenvalue Approximation. Num. Math. 61 (1992), 145-152.
• [2] U. Banerjee, J. E. Osborn, Estimation of the Effect of Numerical Integration in Finite Element Eigenvalue Approximation. Num. Math. 56 (1990), 735-762.
• [3] F. Chatelin, Spectral Approximation of Linear Operators. Computer Science and Applied Mathematics. Academic Press 1983.
• [4] P. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland 1978.
• [5] M. Crouzeix, J. Rappaz, On Numerical Approximation in Bifurcation Theory. Masson, Springer Verlag 1990.
• [6] G. J. Fix, Effects of Quadrature Errors in FE Approx of Steady State, Eigenvalue and Parabolic Problems. The Mathematical Foundations of the FEM with Applications to PDEs (ed. Aziz A.K.). Academic Press 1972, 525-557.
• [7] A. Schatz, A Weak Discrete Maximum Principle and Stability of the FEM in L ͚ on Plane Polygonal Domains. Math. Comp. 34 (1980), 77-91.
• [8] M. Vanmaele, R. Van Keer, Convergence and Error Estimates for a FEM with Numerical Quadrature for a Second Order Elliptic Eigenvalue Problem. Numerical Treatment of Eigenvalue Problems. Vol. 5, eds.: Albrecht J., Collatz L., Hagedorn P., Vette W. International Series of Numerical Mathematics, 96. Birkhäuser Verlag Basel 1991, 225-236.
• [9] M. Vanmaele, R. Van Keer, Error Estimates for a FEM with Numerical Quadrature for a Class of Elliptic Eigenvalue Problems. Numerical Methods, eds.: Greenspan D., Rózsa P. North Holland 1991, 267-282.