Pointwise convergence of some boundary element methods. Part II

Rolf Rannacher; Wolfgang L. Wendland

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1988)

  • Volume: 22, Issue: 2, page 343-362
  • ISSN: 0764-583X

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Rannacher, Rolf, and Wendland, Wolfgang L.. "Pointwise convergence of some boundary element methods. Part II." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 22.2 (1988): 343-362. <http://eudml.org/doc/193533>.

@article{Rannacher1988,
author = {Rannacher, Rolf, Wendland, Wolfgang L.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {boundary element method; convergence estimates; uniform convergence; strongly elliptic boundary integro-differential equations; finite element Galerkin method; pointwise convergence estimates; spline collocation boundary elements},
language = {eng},
number = {2},
pages = {343-362},
publisher = {Dunod},
title = {Pointwise convergence of some boundary element methods. Part II},
url = {http://eudml.org/doc/193533},
volume = {22},
year = {1988},
}

TY - JOUR
AU - Rannacher, Rolf
AU - Wendland, Wolfgang L.
TI - Pointwise convergence of some boundary element methods. Part II
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1988
PB - Dunod
VL - 22
IS - 2
SP - 343
EP - 362
LA - eng
KW - boundary element method; convergence estimates; uniform convergence; strongly elliptic boundary integro-differential equations; finite element Galerkin method; pointwise convergence estimates; spline collocation boundary elements
UR - http://eudml.org/doc/193533
ER -

References

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  1. [1] M. S. BOU EL SEOUD, Kollokationsmethode für schwach singuläre Inté-gralgleichungen erster Art. Z. Angew. Math. Mech. 59, T45-T47 (1979). Zbl0412.65062
  2. [2] M. S. AGRANOVICH, Spectral properties of diffraction problems. In : The General Method of Natural Vibrations in Diffraction Theory. (Russian)(N. N. Voitovic, K. Z. Katzenellenbaum and A. N. Sivov) Izdat. Nauka, Mos-cow 1977. MR484012
  3. [3] M. A. ALEKSIDZE, The Solution of Boundary Value Problems with the Method of Expansion with Respect to Nonorthonormal Functions. Nauka, Moscow 1978 (Russian). MR527813
  4. [4] D. N. ARNOLD and W. L. WENDLAND, On the asymptotic convergence of collocation methods. Math. Comp. 41, 349-381 (1983). Zbl0541.65075MR717691
  5. [5] D. N. ARNOLD and W. L. WENDLAND, The convergence of spline collocationfor strongly elliptic equations on curves. Numer. Math. 47, 317-341 (1985). Zbl0592.65077MR808553
  6. [6] I. BABUSKA and A. K. ziz, Survey lectures on the mathematical foundations of finite element method. In The Mathematical Foundations o f the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), pp. 3-359, Academic Press, New York 1972. Zbl0268.65052MR421106
  7. [7] M. COSTABEL and E. STEPHAN, The normal derivative of the double layer potential on polygons and Galerkin approximation. Applicable Anal. 16, 205-288 (1983). Zbl0508.31003MR712733
  8. [8] COSTABEL and E. STEPHAN, A direct boundary integral equation method for transmission problems. J. Appl. Anal. Appl. 106, 367-413 (1985). Zbl0597.35021MR782799
  9. [9] M. COSTABEL and W. L. WENDLAND, Strong ellipticity of boundary integral operators. J. Reine Angew. Math. 372, 34-63 (1986). Zbl0628.35027MR863517
  10. [10] P. FILIPPI, Layer potentials and acoustic diffraction. J. Sound and Vibration 54, 473-500 (1977). Zbl0368.76073
  11. [11] J. FREHSE and R. RANNACHER, Eine L'-Fehlerabschätzung für diskrete Grundlösungen in der Mehtode der finiten Elemente. In : Finite Elemente, Tagungsband Bonn. Math. Schr. 89, 92-114 (1976). Zbl0359.65093MR471370
  12. [12] J. GIROIRE and J. C. NEDELEC, Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp. 32, 973-990 (1978). Zbl0405.65060MR495015
  13. [13] T. HA DUONG, A finite element method for the double layer potential solutions of the Neumann exterior problem. Math. Meth. Appl. Sci. 2, 191-208 (1980). Zbl0437.65083MR570403
  14. [14] F. K. HEBEKER, An integral equation of the first kind for a free boundary value problem of the stationary Stokes equations. Math. Meth. Appl. Sci. 9, 550-575 (1987). Zbl0656.76031MR1200365
  15. [15] L. HORMANDER, Pseudo-differential operators and non-elliptic boundary problems. Annals Math. 83, 129-209 (1966). Zbl0132.07402MR233064
  16. [16] H. P. HOIDIN, Die Kollokationsmethode angewandt auf die Symmsche Integralgleichung. Doctoral Thesis, EHT Zürich, Switzerland 1983. Zbl0579.65142
  17. [17] G. C. HSIA and W. L. WENDLAND, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449-481 (1977). Zbl0352.45016MR461963
  18. [18] G. C. HSIAO and W. L. WENDLAND, The Aubin-Nitsche lemma for integral equations. Journal of Integral Equations 3, 299-315 (1981). Zbl0478.45004MR634453
  19. [19] F. NATTERER, Über die punktweise Konvergenz finiter Elemente. Number. Math. 25, 67-77 (1975). Zbl0331.65073MR474884
  20. [20] J. C. NEDELEC, Approximation des équations intégrales en mécanique et en physique. Lecture Notes, Centre de Math. Appl. Ecole Polytechnique, 91128 Palaiseau, France, 1977. 
  21. [21] J. C. NEDELEC, Approximation par potentiel de double couche du problème de Neumann extérieur. C. R. Acad. Sci. Paris, Ser. A 286, 616-619 (1978). Zbl0375.65047MR477403
  22. [22] J. C. NEDELEC, Integral equations with non integrable kernels. Integral Equations Operator Theory 5, 562-572 (1982). Zbl0479.65060MR665149
  23. [23] J. A. NITSCHE, L -convergence of finite element approximation. Second Conference on Finite Elements, Rennes, France, 1975. Zbl0362.65088MR568857
  24. [24] P. M. PRENTER, Splines and Variational Methods. John Wiley & Sons, New York 1975. Zbl0344.65044MR483270
  25. [25] S. PRÖSSDORF and G. SCHMIDT, A finite elementcollocation method for singular integral equations. Math. Nachr, 100, 33-60 (1981). Zbl0543.65089MR632620
  26. [26] R. RANNACHER, Punktweise Konvergenz der Methode der finiten Elemente beim Plattenproblem. Manuscripta Math. 19, 401-416 (1976). Zbl0383.65061MR423841
  27. [27] R. RANNACHER, On non conforming and mixed finite element methods for plate bending problems. The linear case. R.A.I.R.O. Anal. Numér. 13, 369-387 (1979). Zbl0425.35042MR555385
  28. [28] R. RANNACHER and R. SCOTT, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38, 437-445 (1982). Zbl0483.65007MR645661
  29. [29] R. RANNACHER and W. L. WENDLAND, On the order of pointwise convergence of some boundary element methods. Part I. Operators of negative and zero order. Math. Modelling and Numer. Analysis 19, 65-88 (1985). Zbl0579.65147MR813689
  30. [30] A. H. SCHATZ and L. B. WAHLBIN, Maximum norm error estimates in the finite element method for Poisson's equation on plante domains with corner. Math. Comp. 32, 73-109 (1978). Zbl0382.65058MR502065
  31. [31] G. SCHMIDT, The convergence of Galerkin and collocation methods with splines for pseudodifferential equations on closed curves. Z. Anal. Anwendungen 3, 371-384 (1984). Zbl0551.65077MR780180
  32. [32] R. SCOTT, Optimal L -estimatees for the finite element method on irregular meshes. Math. Comp. 30, 681-697 (1976). Zbl0349.65060MR436617
  33. [33] E. STEPHAN and W. L. WENDLAND, Remarks to Galerkin and least squares methods with finite elements for general elliptic problems. Manuscripta Geodaetica 1, 93-123 (1976) and Springer Lecture Notes in Math. 564, 461-471 (1976). Zbl0353.65067MR520343
  34. [34] G. STRANG, Approximation in the finite element method. Num. Math. 19 (1972). Zbl0221.65174MR305547
  35. [35] M. E. TAYLOR, Pseudodifferential Operators. Princeton University Press,Princeton, New Jersey 1981. Zbl0453.47026MR618463
  36. [36] F. TREVES, Pseudodifferential Operators. Plenum Press, New York, London 1980. MR597144
  37. [37] J. O. WATSON, Advanced implementation of the boundary element method fortwo- and three-dimensional elastostatics. In : Developments in Boundary Element Methods. I. Banerjee and R. Butterfield (eds.), Appl. Sciences Publ. LTD, London, 31-63 (1979). Zbl0451.73075
  38. [38] W. L. WENDLAND, Boundary element methods and their asymptotic conver-gence. . In : Theoretical Acoustics and Numerical techniques. P. Filippi (éd.), CISM Courses and Lectures No.277, Springer-Verlag, Wien, New York, 135-216 (1983). Zbl0618.65109MR762829

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