Abstract
We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT n. We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operatorA we compute a numerical symbolα C, resp.α G, depending onA and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.
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References
Agranovich, M. S.: On elliptic pseudodifferential operators on a closed curve.Trans. Moscow Math. Soc. 47, 23–74 (1985).
Anderssen, R. S., de Hoog, F. R., Lukas, M. A.:The Application and Numerical Solution of Integral Equations. Sijthoff and Noordhoff, Alphen aan den Rijn 1980.
Arnold, D.N., Saranen, J.: On the asymptotic convergence of spline collocation methods for partial differential equations.SIAM J. Numer. Anal. 21, 459–472 (1984).
Aubin, J.:Approximation of elliptic boundary value problems. Pure and Applied Mathematics. Wiley, New York 1972.
Arnold, D. N., Wendland, W. L.: On the asymptotic convergence of collocation methods.Math. Comp. 41, 349–381 (1983).
Arnold, D. N., Wendland, W. L.: The convergence of spline collocation for strongly elliptic equations on curves.Numer. Math. 47, 317–341 (1985).
Atkinson, K. E.: A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions. Preprint AM 88/8. School of Math., University of New South Wales, Sydney 1988.
Atkinson, K. E.: Piecewise polynomial collocation for integral equations on surfaces in three dimensions.J. Integral Equations 9 ((Suppl.)), 25–48 (1985).
Atkinson, K. E.:A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. SIAM, Philadelphia 1976.
Baker, C. T. H.:The Numerical Treatment of Integral Equations. Clarendon Press, Oxford 1977.
Bramble, J. H., Hilbert, S.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation.SIAM Journal Numerical Analysis 7(1), 112–124 (1970).
Böttcher, A., Silbermann, B.: The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols.Math. Nachr. 110, 279–291 (1983).
Brebbia, C., Kuhn, G., Wendland, W.:Boundary Elements IX. Springer Verlag, Berlin et al. 1987.
Brebbia, C., Telles, J., Wrobel, L.:Boundary Element Techniques. Springer Verlag, Berlin et al. 1984.
Chandler, G., Sloan, I.: Spline qualocation methods for boundary integral equations. Univ. of New South Wales, Preprint 1989.
Ciarlet, P.:The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, New York, Oxford 1978.
Costabel, M.: Principles of boundary element methods.Comp. Phys. Reports 6, 243–274 (1987).
Costabel, M., McLean, W.: Spline collocation for strongly elliptic equations on the torus. (in preparation).
Costabel, M., Penzel, F., Schneider, R.: A collocation method for screen problems inIR 3. To appear in Tagungsberichte “Elliptic Operators on Singular Manifolds”, B.G. Teubner, Leipzig 1991, Breitenbrunn, 30.4.–4.5. 1990, Manuscript, 8 p.
Costabel, M., Penzel, F., Schneider, R.: Error analysis of a boundary element collocation method for a screen problem inIR 3. THD, FB-Math Preprint Nr.1284, 14 p. Febr., THD Darmstadt 1990.
Costabel, M., Stephan, E. P.: Duality estimates for the numerical solution of integral equations.Numer. Math. 54, 339–353 (1989).
Costabel, M., Stephan, E. P.: On the convergence of collocation methods for boundary integral equations on polygons.Math. Comp. 49, 451–478 (1987).
Costabel, M., Wendland, W.: Strong ellipticity of boundary integral operators.J. Reine Angew. Math. 372, 39–63 (1986).
Crouch, S., Starfield, A.:Boundary Element Methods in Solid Mechanics, George Allen & Unwin, London 1983.
Elschner, J.: The double layer potential operator over polyhedral domains II: Galerkin methods.Math. Meth. Appl. Sci. (submitted).
Eskin, G.:Boundary Value Problems for Elliptic Pseudodifferential Equations, vol. 52 ofTranslation of Mathematical Monographs. AMS Providence, Rhode Island 1981.
Filippi, P.:Theoretical Acoustics and Numerical Techniques. CISM Courses and Lectures 277. Springer Verlag, Berlin et al. 1983.
Frank, L.: Difference operators in convolutions.Soviet Math. Dokl. 9, 831–834 (1968).
Frank, L.: Algèbre des opérateurs aux différences finies.Israel J. Math. 13, 24–55 (1972).
Frank, L.: Opérateurs aux différences finies elliptiques.J. Math. Anal. Appl. 45, 260–273 (1974).
Gohberg, I., Feldman, I.:Faltungsgleichungen und Projektionsverfahren zu ihrer Lösung, vol. 49 ofMathem. Reihe. Birkhäuser Verlag, Basel-Stuttgart 1974.
Gohberg, I., Krupnik, N.:Einführung in die Theorie der eindimensionalen singulären Integraloperatoren. Birkhäuser Verlag, Basel, Boston, Stuttgart 1979.
Hackbusch, W.:Integralgleichungen. Theorie und Numerik. Teubner, Stuttgart 1989.
Hagen, R., Silbermann, B.: On finite element collocation for bisingular integral equations.Appl. Anal. 19, 117–135 (1985).
Hagen, R., Silbermann, B.: A Banach algebra approach to the stability of projection methods for singular integral equations.Math. Nachr. 140, 285–297 (1989).
Hörmander, L.:The Analysis of Linear Partial Differential Operators, vol. 3 ofGrundlehren series. Springer Verlag, Berlin, Heidelberg, New York, Tokio 1985.
Hörmander, L.:The Analysis of Linear Partial Differential Operators, vol. 1 ofGrundlehren series. Springer Verlag, Berlin, Heidelberg, New York, Tokio 1985.
Hsiao, G., Prössdorf, S.: A generalization of the Arnold-Wendland lemma to collocation methods for boundary integral equations inIR n. To appear inMath. Meth. Appl. Sci..
Hsiao, G., Wendland, W.: The Aubin-Nitsche lemma for integral equations.J. Integral Equations 3, 299–315 (1981).
Ivanov, V.:The Theory of Approximate Methods and their Application to the Numerical Solution of Singular Integral Equations. Noordhoff, Leyden 1976.
Junghanns, P., Silbermann, B.: Local theory of the collocation method for the approximate solution of singular integral equations.Integral Equations and Operator Theory 7, 791–807 (1984).
Kohn, J., Nirenberg, L.: On the algebra of pseudo-differential operators.Comm. Pure Appl. Math. 18, 269–305 (1965).
Kozak, A. V.: A local principle in the theory of projection methods.Sov. Math. Dokl. 14, 1580–1583 (1973).
Kress, R.:Linear Integral Equations. Springer Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hongkong 1989.
Kumano-go, H.:Pseudodifferential Operators, MIT-Press, Boston 1981.
McLean, W.: Local and global description of periodic pseudodifferential operators. Manuscript, Sydney 1989, to appear in Math. Nachr.
McLean, W.: Periodic pseudodifferential operators and periodic function spaces. Technical report, Univ. of New South Wales, Australia 1989.
Mikhlin, S., Prössdorf, S.:Singular Integral Operators. Springer Verlag, Berlin, Heidelberg, New York, Tokio 1986.
Mukherjee, S.:Boundary Element Methods in Creep and Fracture. Appl. Sci. Publ., London, New York 1982.
Prössdorf, S.: On the super-approximation property of Galerkin's method with finite elements. to appear in Numer. Math.
Prössdorf, S.: Ein Lokalisierungsprinzip in der Theorie der Splineapproximationen und einige Anwendungen.Math. Nachr. 119, 239–255 (1984).
Prössdorf, S.: Numerische Behandlung singulärer Integralgleichungen.Z. angew. Math. Mech. 69, T5-T13 (1989).
Prössdorf, S., Rathsfeld, A.: A spline collocation method for singular integral equations with piecewise continuous coefficients.Int. Eqs. and Op. Th. 7, 536–560 (1984).
Prössdorf, S., Rathsfeld, A.: Stabilitätskriterien für Näherungsverfahren bei singulären Integralgleichungen inL p.Z. Anal. Anw. 6, 539–558 (1987).
Prössdorf, S., Rathsfeld, A.: Mellin techniques in the numerical analysis for one-dimensional singular integral equations.Report R-MATH-06/88, Karl-Weierstrass-Institut, Berlin (1988).
Prössdorf, S., Schmidt, G.: A finite element collocation method for singular integral equations.Math. Nachr. 100, 33–60 (1981).
Prössdorf, S., Schneider, R.: A spline collocation method for multidimensional strongly elliptic pseudodifferential operators of order zero. Fb-math.-perprint no. 1310 “THD Darmstadt May 1990.
Prössdorf, S., Silbermann, B.:Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen. Teubner Texte zur Mathematik. B. G. Teubner, Leipzig 1977.
Prössdorf, S., Silbermann, B.:Numerical Analysis for Integral and Related Operator Equations. Birkhäuser Verlag, Basel, Stuttgart 1990. (to appear).
Rathsfeld, A.: The invertibility of the double layer potential operator in the space of continuous functions defined on a polyhedron. Preprint P-MATH-01/90, Karl-Weierstrass-Institut, Berlin 1990.
Rathsfeld, A.: The invertibility of the double layer potential operator in the space of continuous functions defined on a polyhedron. The panel method.Complex Variables (to appear).
Roch, S., Silbermann, B.: A symbol calculus for finite sections of singular integral operators with shift and piecewise continuous coefficients.J. Funct. Anal. 78(2), 365–389 (1988).
Saranen, J., Wendland, W.: On the asymptotic convergence of collocation methods with spline functions of even degree.Math. Comp. 45(171), 91–108 (1985).
Schmidt, G.: On spline collocation for singular integral equations.Math. Nachr. 111, 177–196 (1983).
Schmidt, G.: On spline collocation methods for boundary integral equations in the plane.Math. Meth. Appl. Sci. 7, 74–89 (1985).
Schmidt, G.: On ε-collocation for pseudodifferential equations on closed curves.
Schmidt, G. Splines und die näherungsweise Lösung von Pseudodifferentialgleichungen auf geschlossenen Kurven. Report R-MATH-09/86, Karl-Weierstrass Institut, Berlin 1986.
Schneider, R.: Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations.Numer. Math. 58, 855–873 (1991).
Shubin, M.:Pseudodifferential Operators and Spectral Theory, Nauka, Moscow 1978. (Russian, engl. transl. Springer Verlag, 1987).
Silbermann, B.: Symbol constructions and numerical analysis. Preprint, Techn. Univ. Chemnitz, 1989.
Silbermann, B.: Lokale Theorie des Reduktionsverfahrens für singuläre Integragleichungen.Z. Anal. Anw. 1(6), 45–56 (1982).
Silberman, B.: Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren.Math. Nachr. 104, 137–146 (1981).
Sloan, I. H., Wendland, W. L.: A quadrature-based approach to improving the collocation method for splines of even degree, to appear in Zeitschr. Anal. u. Anwend.
Stein, E., Weiss, G.:Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press 1971.
Stephan, E.:Differenzenapproximation von Pseudo-Differenzenoperatoren. PhD thesis, Technische Hochschule Darmstadt 1975.
Taylor, M.:Pseudodifferential Operators. Princeton University Press, Princeton, N.J. 1981.
Vainikko, G.:Funktionalanalysis der Diskretisierungsmethoden. Teubner-Texte zur Mathematik. B. G. Teubner, Leipzig 1976.
Vainikko, G.: Collocation methods for multidimensional weakly singular integral equations. In:Praktische Behandlung von Integralgleichungen, Randelementmethoden und singulären Gleichungen. Mathematisches Forschungsinstitut Oberwolfach, Tagungsberichte 1988.
Vainikko, G., Pedas, A., Uba, P.:Methods for Solving Weakly Singular Integral Equations. Gos. Uni. Tartu 1984. (Russian).
Wendland, W. L.: On some mathematical aspects of boundary element methods for elliptic problems. In:Mathematics of Finite Elements and Applications V (J. Whiteman, ed.), pp. 193–227. Academic Press, London 1985.
Wendland, W. L.: Strongly elliptic boundary integral equations. In:The State of the Art in Numerical Analysis (A. iserles, M. Powell, eds.), pp. 511–562. Clarendon Press, Oxford 1987.
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The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.
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Prössdorf, S., Schneider, R. Spline approximation methods for multidimensional periodic pseudodifferential equations. Integr equ oper theory 15, 626–672 (1992). https://doi.org/10.1007/BF01195782
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DOI: https://doi.org/10.1007/BF01195782