The effect of numerical quadrature in the 𝑝-version of the finite element method

Banerjee, Uday; Suri, Manil

doi:10.1090/S0025-5718-1992-1134712-5

The effect of numerical quadrature in the -version of the finite element method

Authors: Uday Banerjee and Manil Suri
Journal: Math. Comp. 59 (1992), 1-20
MSC: Primary 65D30; Secondary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1992-1134712-5
MathSciNet review: 1134712
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the use of numerical quadrature in the p-version of the finite element method. We describe a set of minimal conditions that the quadrature rules should satisfy, for various types of elements. Under sufficient assumptions of smoothness, we establish optimality of the asymptotic rate of convergence. Some computational results are presented, which illustrate under what conditions overintegration may be useful.

[1] I. Babuška and Manil Suri, The optimal convergence rate of the 𝑝-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776. MR 899702, https://doi.org/10.1137/0724049
[2] I. Babuška, B. Guo, and M. Suri, Implementation of nonhomogeneous Dirichlet boundary conditions in the p-version of the finite element method, Impact Comput. Sci. Engrg. 1 (1989), 36-63.
[3] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86. MR 637287, https://doi.org/10.1090/S0025-5718-1982-0637287-3
[4] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
[5] P. G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 409–474. MR 0421108
[6] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR 0448814
[7] Milo R. Dorr, The approximation theory for the 𝑝-version of the finite element method, SIAM J. Numer. Anal. 21 (1984), no. 6, 1180–1207. MR 765514, https://doi.org/10.1137/0721073
[8] D. A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg. 21 (1985), no. 6, 1129–1148. MR 794241, https://doi.org/10.1002/nme.1620210612
[9] D. A. Dunavant, Economical symmetrical quadrature rules for complete polynomials over a square domain, Internat. J. Numer. Methods Engrg. 21 (1985), no. 10, 1777–1784. MR 809279, https://doi.org/10.1002/nme.1620211004
[10] V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
[11] W. Gui and I. Babuška, The ℎ,𝑝 and ℎ-𝑝 versions of the finite element method in 1 dimension. I. The error analysis of the 𝑝-version, Numer. Math. 49 (1986), no. 6, 577–612. MR 861522, https://doi.org/10.1007/BF01389733
[12] J. N. Lyness, QUG2-integration over a triangle, Technical Memo #13, Math. and Comp. Sci. Div., Argonne National Lab., 1983.
[13] Yvon Maday and Einar M. Rønquist, Optimal error analysis of spectral methods with emphasis on nonconstant coefficients and deformed geometries, Comput. Methods Appl. Mech. Engrg. 80 (1990), no. 1-3, 91–115. Spectral and high order methods for partial differential equations (Como, 1989). MR 1067944, https://doi.org/10.1016/0045-7825(90)90016-F
[14] H.-S. Oh and I. Babuška, The p-version of the finite element method for the elliptic boundary value problems with interfaces, Comput. Methods Appl. Mech. Engrg. (1992) (in press).
[15] Manil Suri, The 𝑝-version of the finite element method for elliptic equations of order 2𝑙, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 2, 265–304 (English, with French summary). MR 1052150, https://doi.org/10.1051/m2an/1990240202651
[16] L. B. Wahlbin, Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration, RAIRO Anal. Numér. 12 (1978), no. 2, 173–202, v (English, with French summary). MR 502070, https://doi.org/10.1051/m2an/1978120201731