Numerical evaluation of surface integrals in three dimensions

Author:
David Chien

Journal:
Math. Comp. **64** (1995), 727-743

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1995-1270615-6

MathSciNet review:
1270615

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the evaluation of surface integrals over piecewise smooth surfaces in three dimensions. The method consists in first replacing a parametrization for the surface and the integrand function by piecewise polynomial interpolants of them, and second, using a numerical integration scheme for the resulting integral. The order of convergence is higher than would be expected based on the underlying interpolation theory.

**[1]**K. Atkinson,*Piecewise polynomial collocation for integral equations on surfaces in three dimensions*, J. Integral Equations**9**(Suppl.), (1985), 25-48. MR**792418 (87g:65161)****[2]**-,*An empirical study of the numerical solution of integral equations on surfaces in*, Reports on Computational Mathematics, #1, Dept. of Mathematics, University of Iowa, Iowa City, IA, 1989.**[3]**-,*User's guide to a boundary element package for solving integral equations on piecewise smooth surfaces*, Reports on Computational Mathematics, #44, Dept. of Mathematics, University of Iowa, Iowa City, IA, 1993.**[4]**K. Atkinson and E. Venturino,*Numerical evaluations of line integrals*, SIAM J. Numer. Anal.**30**(1993), 881-888. MR**1220659 (94b:65033)****[5]**D. Chien,*Piecewise polynomial collocation for integral equations on surfaces in three dimensions*, Ph.D. thesis, Univ. of Iowa, Iowa City, IA, 1991.**[6]**N. Günter,*Potential theory and its applications to basic problems of mathematical physics*, Ungar, New York, 1967.**[7]**A. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Englewood Cliffs, NJ, 1971. MR**0327006 (48:5348)****[8]**K. Georg and J. Tausch,*Some error estimates for the numerical approximation of surface integrals*, Math. Comp.**62**(1994), 755-763. MR**1219704 (94g:65025)****[9]**J. Lyness,*Quadrature over a simplex*, SIAM J. Numer. Anal.**15**(1978), 122-133 and 870-887.**[10]**-,*Quadrature over curved surfaces by extrapolation*, Math. Comp.**63**(1994), 727-740. MR**1257576 (95a:65050)****[11]**C. Schwab and W. Wendland,*On numerical cubatures of singular surface integrals in boundary element methods*, Numer. Math.**62**(1992), 343-369. MR**1169009 (93h:65035)****[12]**P. Verlinden and R. Cools,*Proof of a conjectured asymptotic expansion for the approximation of surface integrals*, Math. Comp.**63**(1994), 717-725. MR**1257581 (95a:65047)****[13]**Y. Yang and K. Atkinson,*Numerical integration for multivariable functions with point singularities*, Reports on Computational Mathematics, #41, Dept of Mathematics, University of Iowa, Iowa City, IA, 1993.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1995-1270615-6

Article copyright:
© Copyright 1995
American Mathematical Society