Product integration-collocation methods for noncompact integral operator equations

Authors:
G. A. Chandler and I. G. Graham

Journal:
Math. Comp. **50** (1988), 125-138

MSC:
Primary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917821-1

MathSciNet review:
917821

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact. Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary. We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions. These are sufficient to ensure stability, but do not damage the rate of convergence. Numerical experiments show that such modifications are necessary in certain circumstances.

**[1]**K. E. Atkinson,*A Survey of Numerical Methods for the Solution of Integral Equations of the Second Kind*, SIAM, Philadelphia, Pa., 1976.**[2]**K. Atkinson and F. de Hoog,*The numerical solution of Laplace’s equation on a wedge*, IMA J. Numer. Anal.**4**(1984), no. 1, 19–41. MR**740782**, https://doi.org/10.1093/imanum/4.1.19**[3]**Carl de Boor,*Good approximation by splines with variable knots*, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Birkhäuser, Basel, 1973, pp. 57–72. Internat. Ser. Numer. Math., Vol. 21. MR**0403169****[4]**Carl de Boor,*A practical guide to splines*, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR**507062****[5]**C. A. Brebbia, J. C. F. Telles & L. C. Wrobel,*Boundary Element Techniques*, Springer-Verlag, Berlin and New York, 1984.**[6]**H. G. Burchard,*On the degree of convergence of piecewise polynomial approximation on optimal meshes*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 531–559. MR**0481758**, https://doi.org/10.1090/S0002-9947-1977-0481758-4**[7]**G. A. Chandler,*Superconvergence of Numerical Solutions to Second Kind Integral Equations*, Thesis, Australian National University, 1979.**[8]**G. A. Chandler,*Galerkin’s method for boundary integral equations on polygonal domains*, J. Austral. Math. Soc. Ser. B**26**(1984), no. 1, 1–13. MR**750551**, https://doi.org/10.1017/S033427000000429X**[9]**G. A. Chandler,*Superconvergent approximations to the solution of a boundary integral equation on polygonal domains*, SIAM J. Numer. Anal.**23**(1986), no. 6, 1214–1229. MR**865952**, https://doi.org/10.1137/0723082**[10]**G. A. Chandler & I. G. Graham,*Product Integration-Collocation Methods for Non-Compact Integral Operator Equations*, Research Report CMA-R41-85, Centre for Mathematical Analysis, Australian National University, Canberra, 1985.**[11]**Martin Costabel and Ernst Stephan,*Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation*, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR**874845****[12]**Martin Costabel,*Boundary integral operators on curved polygons*, Ann. Mat. Pura Appl. (4)**133**(1983), 305–326. MR**725031**, https://doi.org/10.1007/BF01766023**[13]**M. L. Dow and David Elliott,*The numerical solution of singular integral equations over (-1,1)*, SIAM J. Numer. Anal.**16**(1979), no. 1, 115–134. MR**518688**, https://doi.org/10.1137/0716009**[14]**R. De Vore & K. Scherer, "Variable knot, variable degree spline approximation to ," in*Quantitative Approximation*(R. De Vore and K. Scherer, eds.), Academic Press, New York, 1980, pp. 121-131.**[15]**I. G. Graham and G. A. Chandler,*High-order methods for linear functionals of solutions of second kind integral equations*, SIAM J. Numer. Anal.**25**(1988), no. 5, 1118–1137. MR**960869**, https://doi.org/10.1137/0725064**[16]**M. A. Jaswon & G. I. Symm,*Integral Equation Methods in Potential Theory and Electrostatics*, Academic Press, New York, 1977.**[17]**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****[18]**M. G. Krein,*Integral Equations on a Half Line with Kernel Depending Upon the Difference of the Arguments*, Amer. Math. Soc. Transl., vol. 22, Amer. Math. Soc., Providence, R.I., 1963, pp. 163-288.**[19]**U. Lamp, T. Schleicher, E. Stephan, and W. L. Wendland,*Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem*, Computing**33**(1984), no. 3-4, 269–296 (English, with German summary). MR**773929**, https://doi.org/10.1007/BF02242273**[20]**W. McLean,*Boundary Integral Methods for the Laplace Equation*, Thesis, Australian National University, Canberra, 1985.**[21]**John R. Rice,*On the degree of convergence of nonlinear spline approximation*, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 349–365. MR**0267324****[22]**Claus Schneider,*Product integration for weakly singular integral equations*, Math. Comp.**36**(1981), no. 153, 207–213. MR**595053**, https://doi.org/10.1090/S0025-5718-1981-0595053-0**[23]**Ian H. Sloan,*A review of numerical methods for Fredholm equations of the second kind*, Application and numerical solution of integral equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978) Monographs Textbooks Mech. Solids Fluids: Mech. Anal., vol. 6, Nijhoff, The Hague, 1980, pp. 51–74. MR**582984****[24]**I. H. Sloan and A. Spence,*Projection methods for integral equations on the half-line*, IMA J. Numer. Anal.**6**(1986), no. 2, 153–172. MR**967661**, https://doi.org/10.1093/imanum/6.2.153**[25]**I. N. Sneddon & S. C. Das, "The stress intensity factor at the tip of an edge crack in an elastic half plane,"*Internat. J. Engrg. Sci.*, v. 9, 1971, pp. 25-36.**[26]**I. N. Sneddon and M. Lowengrub,*Crack problems in the classical theory of elasticity*, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0258339****[27]**M. P. Stallybrass,*A pressurized crack in the form of a cross*, Quart. J. Mech. Appl. Math.**23**(1970), 35–48. MR**0261843**, https://doi.org/10.1093/qjmam/23.1.35**[28]**M. P. Stallybrass,*A crack perpendicular to an elastic half-plane*, Internat. J. Engrg. Sci.**8**(1970), 351–362 (English, with French, German, Italian and Russian summaries). MR**0261842**, https://doi.org/10.1016/0020-7225(70)90073-X**[29]**Frank Stenger,*Numerical methods based on Whittaker cardinal, or sinc functions*, SIAM Rev.**23**(1981), no. 2, 165–224. MR**618638**, https://doi.org/10.1137/1023037**[30]**W. L. Wendland,*On some mathematical aspects of boundary element methods for elliptic problems*, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 193–227. MR**811035**

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20

Retrieve articles in all journals with MSC: 65R20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917821-1

Keywords:
Second-kind integral equations,
product integration,
boundary integral equations,
collocation

Article copyright:
© Copyright 1988
American Mathematical Society