Abstract
In this paper, we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k, in the context of approximating linear functionals, which are also known as “quantities of interest”. We have obtained the optimal order of convergence, \({\mathcal{O}(h^{2k})}\), of the error in the computed functional, when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k − 1. However, this result was obtained under an increased regularity assumption on the data, which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature. We have obtained a lower bound of the error in the computed functional for a particular problem, which indicates the necessity of the increased regularity requirement of the data. Numerical experiments have been presented indicating that over-integration may be necessary to accurately approximate the functional, when the data lack the increased regularity.
Similar content being viewed by others
References
Babuška I., Miller A.: The post-processing approach in the finite element method. Part I: calculation of displacements, stresses and higher order derivatives of displacement. Int. J. Numer. Methods Eng. 20, 1085–1109 (1984)
Babuška I., Miller A.: The post-processing approach in the finite element method. Part II: calculation of stress intensity factors. Int. J. Numer. Methods Eng. 20, 1111–1129 (1984)
Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol. II, pp. 641–787. North-Holland, Amsterdam (1991)
Babuška I., Strouboulis T.: The Finite Element Method and Its Reliability. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2001)
Babuška I., Aziz A.K.: Survey lectures on the mathematical foundation of the finite element method. In: Aziz, A.K. (ed) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 3–359. Academic Press, New York (1972)
Banerjee U., Osborn J.E.: Estimation of the effect of numerical integration in finite element eigenvalue approximation. Numer. Math. 56, 735–762 (1990)
Banerjee U., Suri M.: The effect of numeical quadrature in the p-version of the finite element method. Math. Comput. 59, 1–20 (1992)
Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics. Springer, New York (2002)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North-Holland, Amsterdam (1978)
Ciarlet P.G., Raviart P.-A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In: Aziz, A.K. (ed) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 404–474. Academic Press, New York (1972)
Fix G.J.: Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems. In: Aziz, A.K. (ed) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 525–556. Academic Press, New York (1972)
Herbold R.J., Schultz M.H., Varga R.S.: The effect of quadrature errors in the numerical solution of boundary value problems by variational techniques. Aequ. Math. 3, 247–270 (1969)
Herbold R.J., Varga R.S.: The effect of quadrature errors in the numerical solution of two-dimensional boundary value problems by variational techniques. Aeq. Math. 7, 36–58 (1972)
Schatz A.H., Wahlbin L.B.: Interior maximum-norm estimates for finite element methods, part II. Math. Comput. 64(221), 907–928 (1995)
Szabó B., Babuška I.: Finite Element Analysis. Wiley, New York (1991)
Wahlbin L.B.: Superconvergence in the Galerkin Finite Element Methods. Lecture Notes in Mathematics, vol. 1605. Springer, New York (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of I. Babuška was partially supported by NSF grant DMS 0611094. The research of U. Banerjee was partially supported by NSF grant DMS 0610778.
Rights and permissions
About this article
Cite this article
Babuška, I., Banerjee, U. & Li, H. The effect of numerical integration on the finite element approximation of linear functionals. Numer. Math. 117, 65–88 (2011). https://doi.org/10.1007/s00211-010-0335-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-010-0335-2