The convergence of spline collocation for strongly elliptic equations on curves

Douglas N. Arnold, Wolfgang L. Wendland

Research output: Contribution to journalArticle

108 Citations (Scopus)

Abstract

Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.

Original languageEnglish (US)
Pages (from-to)317-341
Number of pages25
JournalNumerische Mathematik
Volume47
Issue number3
DOIs
StatePublished - Sep 1 1985

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Spline Collocation
Splines
Elliptic Equations
Spline
Midpoint
Collocation
Curve
Odd
Panel Method
Graded Meshes
Integro-differential Operators
Cauchy Integral
Asymptotic Convergence
Fourier Expansion
Constant function
Ellipticity
Boundary Integral
Singular Integral Equation
Boundary element method
Variable Coefficients

Keywords

  • Subject Classifications: AMS(MOS): 65R20, 65N99, 65N30, 65E05, 30C30, 73K30, 65N35, CR: G. 1.9

Cite this

The convergence of spline collocation for strongly elliptic equations on curves. / Arnold, Douglas N.; Wendland, Wolfgang L.

In: Numerische Mathematik, Vol. 47, No. 3, 01.09.1985, p. 317-341.

Research output: Contribution to journalArticle

Arnold, Douglas N. ; Wendland, Wolfgang L. / The convergence of spline collocation for strongly elliptic equations on curves. In: Numerische Mathematik. 1985 ; Vol. 47, No. 3. pp. 317-341.
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