×

zbMATH — the first resource for mathematics

The convergence of spline collocation for strongly elliptic equations on curves. (English) Zbl 0592.65077
This paper presents a unified asymptotic error analysis for even as well as for odd degree splines subordinate. Uniform or smoothly graded meshes are in consideration only. The asymptotic convergence of optimal order is proved.
The crucial assumption for the generalized boundary integral and integro- differential operators is strong ellipticity. The analysis is based on Fourier expansion, it extends known results to variable coefficient equations. This paper contains the first convergence proof of midpoint collocation with piecewise constant functions.
Reviewer: V.Drápalík

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Agranovitch, M.S.: Spectral properties of elliptic pseudodifferential operators on a closed curve. Russ. Math. Surv.13, 279-281 (1979)
[2] Arnold, D.N.: A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method. Math. Comput.41, 383-397 (1983) · Zbl 0543.65087
[3] Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comput.41, 349-381 (1983) · Zbl 0541.65075
[4] Atkinson, K.E.: A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. Philadelphia: Soc. Ind. Appl. Math. 1976 · Zbl 0353.65069
[5] Atkinson, K.E., Graham, I., Sloan, I.: Piecewise continuous collocation for integral equations. SIAM J. Numer. Anal.20, 172-186 (1983) · Zbl 0514.65094
[6] Banerjee, P.K., Butterfield, R.: Developments in Boundary Element Methods-1. London: Appl. Sci. Publ. 1979 · Zbl 0446.00012
[7] Banerjee, P.K., Shaw, R.P.: Developments in Boundary Element Methods-2. London: Appl. Sci. Publ. 1982 · Zbl 0476.00027
[8] Bolteus, L., Tullberg, O.: BEMSTAT ? A new type of boundary element program for two-dimensional elasticity problems. In: Boundary Element Methods. C.A. Brebbia (ed.), pp. 518-537. Berlin-Heidelberg-New York: Springer 1981 · Zbl 0472.73094
[9] Brebbia, C.A.: Progress in Boundary Element Methods. London, Plymouth: Pentech Press 1981 · Zbl 0475.73083
[10] Brebbia, C.A.: Boundary Element Methods. Berlin-Heidelberg-New York: Springer 1981 · Zbl 0475.73083
[11] Brebbia, C.A.: Boundary Element Techniques, Methods in Engineering. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0528.65058
[12] Brebbia, C.A., Futagami, T., Tanaka, M.: Boundary Elements. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0521.00024
[13] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0556.73086
[14] Chatelin, F.: Spectral Approximation of Linear Operators. New York: Academic Press 1983 · Zbl 0517.65036
[15] Crouch, S.L., Starfield, A.H.: Boundary Element Methods in Solid Mechanics. London: George Allen & Unwin 1983 · Zbl 0528.73083
[16] Cruse, T.A.: Application of the boundary-integral equation solution method in solid mechanics. In: Variational Methods in Engineering. Dept. Civil Eng. Southampton Univ., England, 9.1-9.29, 1972
[17] Douglas, J., Jr., Dupont, T., Wahlbin, L.: OptimalL ? error estimates for Galerkin approximations to solutions of two point boundary value problems. Math. Comput.29, 475-583 (1975)
[18] Elschner, J., Schmidt, G.: On spline interpolation in periodic Sobolev spaces. preprint P-Math-01/83. Institut für Mathematik, Akademie der Wissenschaften der DDR, Berlin 1983 · Zbl 0601.41016
[19] Filippi, P.: Theoretical Acoustics and Numerical Techniques. CISM Courses and Lectures 277. Wien-New York: Springer 1983 · Zbl 0544.00039
[20] Hämmerlin, G., Schumaker, L.L.: Procedures for kernel approximation and solution of Fredholm integral equations of the second kind. Numer. Math.34, 125-141 (1980) · Zbl 0424.65065
[21] Hayes, J.K., Kahaner, D.K., Kellner, R.G.: An improved method for numerical conformal mapping. Math. Comput.26, 327-334 (1972) · Zbl 0239.65033
[22] Helfrich, H.-P.: Simultaneous approximation in negative norms of arbitrary order. R.A.I.R.O. Numer. Anal.15, 231-235 (1981) · Zbl 0495.41010
[23] Hörmander, L.: Fourier integral operators I. Acta Math.127, 79-183 (1971) · Zbl 0212.46601
[24] Hoidn, H.-P.: Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung. Doctoral Thesis ETH Zürich, Switzerland, 1983 · Zbl 0579.65142
[25] Hsiao, G.C., Kopp, P., Wendland, W.L.: Some applications of a Galerkin collocation method for integral equations of the first kind. Math. Meth. Appl. Sci.6, 280-325 (1984) · Zbl 0546.65091
[26] Ivanov, V.V.: The Theory of Approximate Methods and their Application to the Numerical Solution of Singular Integral Equations. Leyden: Noordhoff Int. Publ. 1976 · Zbl 0346.65065
[27] Mukherjee, S.: Boundary Element Methods in Creep and Fracture. London-New York: Applied Sci. Publ. 1982 · Zbl 0534.73070
[28] Mustoe, G.G., Mathews, I.C.: Direct boundary integral methods, point collocation and variational procedures (To appear)
[29] Nitsche, J., Schatz, A.: On local approximation properties ofL 2-projections on spline subspaces. Appl. Anal.2, 161-168 (1972) · Zbl 0239.41007
[30] Nitsche, J., Schatz, A.: Interior estimates for Ritz-Galerkin methods. Math. Comput.28, 937-958 (1974) · Zbl 0298.65071
[31] Noble, B.: Error analysis of collocation methods for solving Fredholm integral equations. In: Topics in Numerical Analysis. J.H. Miller (ed.), pp. 211-232. London: Academic Press 1972
[32] Prenter, P.M.: A collocation method for the numerical solution of integral equations. SIAM J. Numer. Anal.10, 570-581 (1973) · Zbl 0261.65085
[33] Prössdorf, S.: Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen. Math. Nachr.119, 239-255 (1984) · Zbl 0601.65033
[34] Prössdorf, S., Rathsfeld, A.: A spline collocation method for singular integral equations with piecewise continuous coefficients. Integral Equations Oper. Theory7, 536-560 (1984) · Zbl 0573.65106
[35] Prössdorf, S., Rathsfeld, A.: On spline Galerkin methods for singular integral equations with piecewise continuous coefficients. (To appear in Numer. Math.) · Zbl 0686.65096
[36] Prössdorf, S., Schmidt, G.: A finite element collocation method for singular integral equations. Math. Nachr.100, 33-60 (1981) · Zbl 0543.65089
[37] Prössdorf, S., Schmidt, G.: A finite element collocation method for systems of singular integral equations. Preprint P-MATH-26/81. Institut für Mathematik, Akademie der Wissenschaften der DDR. Berlin 1981 · Zbl 0464.65092
[38] Rizzo, F.J.: An integral equation approach to boundary value problems of classical elastostatics. Quart. Appl. Math.25, 83-95 (1967) · Zbl 0158.43406
[39] Saranen, J., Wendland, W.L.: On the asymptotic convergence of collocation methods with spline functions of even degree. (Preprint 700, Math., Techn. Univ. Darmstadt 1982) (To appear in Math. Comput.45 (1985)) · Zbl 0623.65145
[40] Saranen, J., Wendland, W.L.: The Fourier series representation of pseudodifferential operators on closed curves. (In preparation) · Zbl 0577.47046
[41] Schmidt, G.: On spline collocation for singular integral equations. Math. Nachr.111, 177-196 (1983) · Zbl 0543.65088
[42] Schmidt, G.: On spline collocation methods for boundary integral equations in the plane. (To appear in Math. Meth. Appl. Sci.7 (1985)) · Zbl 0577.65107
[43] Schmidt, G.: The convergence of Galerkin and collocation methods with splines for pseudodifferential equations on closed curves. Z. Anal. Anw.3, 371-384 (1984) · Zbl 0551.65077
[44] Seeley, R.: Topics in pseudodifferential operators. In: Pseudo-Differential Operators. L. Nirenberg (ed.), pp. 169-305. Rome: Edizione Cremonese 1969
[45] Strichartz, R.S.: Multipliers on fractional Sobolev spaces. J. Math. Mech.16, 1031-1061 (1967) · Zbl 0145.38301
[46] Symm, G.T.: Integral equation methods in potential theory II. Proc. Royal Soc. London A275, 33-46 (1963) · Zbl 0112.33201
[47] Taylor, M.: Pseudodifferential Operators. Princeton: Princeton University Press 1981 · Zbl 0453.47026
[48] Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators I. New York-London: Plenum Press 1980
[49] Watson, J.O.: Hermitian cubic boundary elements for plane problems of fracture mechanics. Res. Mechanica4, 23-42 (1982)
[50] Wendland, W.L.: Boundary element methods and their asymptotic convergence. In: Theoretical Acoustics and Numerical Techniques. P. Filippi (ed.), pp. 135-216. CISM courses and lectures 277. Wien-New York: Springer 1983
[51] Quade, W., Collatz, L.: Zur Interpolationstheorie der reellen periodischen Funktionen. Sonderausgabe d. Sitzungsber. d. Preußischen Akad. d. Wiss., Phys.-math. Kl., pp. 1-49. Berlin: Verlag d. Akad. d. Wiss. 1938 · JFM 65.0543.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.