Proof of a conjectured asymptotic expansion for the approximation of surface integrals. (English) Zbl 0811.65020
Recently K. Georg [SIAM J. Sci. Stat. Comput. 12, No. 2, 443-453 (1991; Zbl 0722.65005)] introduced modified trapezoidal and midpoint rules to approximate a surface integral over a curved triangular surface, which do not require that the Jacobian be known explicitly, and conjectured that these rules admit an asymptotic expansion in even powers of \(1/n\), where \(n^ 2\) is the number of subtriangles. This paper presents a proof of this conjecture. The same conjecture was established by J. N. Lyness [Quadrature over curved surfaces by extrapolation, Math. Comput. 63, No. 208, 727-740 (1994)] simultaneously and independently.
Reviewer: J.Elschner (Berlin)
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 65B15 | Euler-Maclaurin formula in numerical analysis |
| 41A55 | Approximate quadratures |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 41A63 | Multidimensional problems |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
Keywords:
Euler-Maclaurin expansion; boundary element method; trapezoidal and midpoint rules; surface integral; curved triangular surface; asymptotic expansionCitations:
Zbl 0722.65005References:
| [1] | Kurt Georg, Approximation of integrals for boundary element methods, SIAM J. Sci. Statist. Comput. 12 (1991), no. 2, 443 – 453. · Zbl 0722.65005 · doi:10.1137/0912024 |
| [2] | Kurt Georg and Johannes Tausch, Some error estimates for the numerical approximation of surface integrals, Math. Comp. 62 (1994), no. 206, 755 – 763. · Zbl 0803.65022 |
| [3] | J. N. Lyness, Quadrature over curved surfaces by extrapolation, Math. Comp. 63 (1994), no. 208, 727 – 740. · Zbl 0812.65011 |
| [4] | J. N. Lyness, Quadrature over a simplex. I. A representation for the integrand function, SIAM J. Numer. Anal. 15 (1978), no. 1, 122 – 133. · Zbl 0388.41016 · doi:10.1137/0715008 |
| [5] | Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0253.46001 |
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