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On the order of pointwise convergence of some boundary element methods. I: Operators of negative and zero order. (English) Zbl 0579.65147
The approximate solution of boundary integral equations and strongly elliptic pseudodifferential equations by the finite element Galerkin method are examined. For operators of order $$2\alpha\leq 0$$ it is shown that the discrete solutions and for the case of some first kind integral equations also the traces of the corresponding potentials converge uniformly with almost the same optimal order as is known for their convergence in the mean square sense. The proof is based on error estimates for discrete Green functions which are derived by using weighted Sobolev norms and Garding’s inequality.
Reviewer: N.F.F.Ebecken

##### MSC:
 65R20 Numerical methods for integral equations 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 45K05 Integro-partial differential equations 35S15 Boundary value problems for PDEs with pseudodifferential operators
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##### References:
 [1] 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 Moscow 1977 MR484012 [2] D ARNOLD and W L WENDLANDOn the asymptotic convergence of collocation methods Math Comp in print (1983) Zbl0541.65075 MR717691 · Zbl 0541.65075 [3] J P AUBIN, Approximation of Elliptic Boundary-Value Problems Wiley-Interscience New York 1972 Zbl0248.65063 MR478662 · Zbl 0248.65063 [4] I BABUSKA and A K Aziz, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A K Aziz ed ) pp 3-359, Academic Press, New York 1972 Zbl0259.00014 MR347104 · Zbl 0259.00014 [5] T DUPONT and R SCOTT, Constructive polynomial approximation In < Recent Advances in Numerical Analysis > (C de Boor ed ), Proc at MRC Madison Wisconsin, May 1978 Zbl0456.65003 · Zbl 0456.65003 [6] G I ESKIN, Boundary Value Problems for Elliptic Pseudodifferential Equations Trans Math Mon Amer Math Soc Providence, Rhode Island 1981 1981 Zbl0458.35002 MR623608 · Zbl 0458.35002 [7] J FREHSE and R RANNACHEREine $$L^1$$-Fehlerabschatzung fur diskrete Grundlosungen in der Methode der finiten Elemente In < Finite Elemente > Tagungsband Bonn Math Schr 89, 92-114 (1976) Zbl0359.65093 MR471370 · Zbl 0359.65093 [8] L HORMANDER, Pseudo-differential operators and non-elliptic boundary problems Annals Math 83, 129-209 (1966) Zbl0132.07402 MR233064 · Zbl 0132.07402 [9] L HORMANDER, Fourier intégral operators I Acta mathematica 127, 79-183 (1971) Zbl0212.46601 MR388463 · Zbl 0212.46601 [10] G C HSIAO, P KOPP and W L WENDLAND, A Galerkin collocation method for some integral equations of the first kind Computing 25, 89-130 (1980) Zbl0419.65088 MR620387 · Zbl 0419.65088 [11] G C HSIAO and W L WENDLAND, Afinite element method for some integral equations of the first kind J Math Anal Appl 58, 449-481 (1977) Zbl0352.45016 MR461963 · Zbl 0352.45016 [12] G C HSIAO and W L WENDLANDThe Aubin-Nitsche lemma for integral equations Journal of Integral Equations 3, 299-315 (1981) Zbl0478.45004 MR634453 · Zbl 0478.45004 [13] G C HSIAO and W L WENDLANDSuper-approximation for boundary integral methods In Advances in Computer Methods for Partial Differential Equations IV (ed R Vichnevetsky, R S Stepleman), pp 200-206, IMACS, Dept Comp Sc Rutgers Univ New Brunswick 1981 [14] E MARTENSENPotentialtheorie B G Teubner Stuttgart1968 Zbl0174.42602 MR247116 · Zbl 0174.42602 [15] S G MICHLIN, Vorlesungen uber lineare Integralgleichungen Verl der Wiss Berlin1962 MR141959 [16] F NATTERER, Uber die punktweise Konvergenz finiter Elemente Numer Math 25 67-77 (1975) Zbl0331.65073 MR474884 · Zbl 0331.65073 [17] J C NEDELEC, Approximation des équations integrales en mecanique et en physique Lecture Notes, Centre de Math Appl Ecole Polytechnique, 91128 Palaiseau, France, 1977 [18] J C NEDELEC and J PLANCHARD, Une methode variationnelle d’elements finis pour la resolution numérique d un probleme exterieur des R3 Revue Franc Automatique, Inf Rech Oper R 3, 105-129 (1973) Zbl0277.65074 MR424022 · Zbl 0277.65074 [19] J A NITSCHE , L$$^\infty$$-convergence of finite element approximation Second Conference on Finite Elements, Rennes, France, 1975 Zbl0362.65088 MR568857 · Zbl 0362.65088 [20] P M PRENTER, Splines and Variational Methods John Wiley & Sons, New York 1975 Zbl0344.65044 MR483270 · Zbl 0344.65044 [21] R RANNACHER, Punktweise Konvergenz der Methode der finiten Elemente beim Plattenproblem Manuscripta math 19, 401-416(1976) Zbl0383.65061 MR423841 · Zbl 0383.65061 [22] J SARANEN and W L WENDLAND, One the asymptotic convergence of collocation methods with spline functions of even degree, to appear in Math Comp 1985 Zbl0623.65145 MR790646 · Zbl 0623.65145 [23] A H SCHATZ and L B WAHLBIN, Maximum norm error estimates in the finite element method for Poisson equation on plane domains with corners Math Comp 32, 73-109 (1978) Zbl0382.65058 MR502065 · Zbl 0382.65058 [24] R SCOTTOptimal $$L^\infty$$-estimates for the finite element method on irregular meshes Math Comp 30, 681-697 (1976) Zbl0349.65060 MR436617 · Zbl 0349.65060 [25] E STEPHAN, Solution procedures for interface problems in acoustics and electro-magnetics In Theoretical Acoustics and Numerical Techniques (ed P Filippi), CISM Courses 277, Springer-Verlag, Wien, New York, 291-348 (1983) Zbl0578.76078 MR762832 · Zbl 0578.76078 [26] E STEPHAN and W L WENDLAND, Remarks to Galerkin and least squares methods with finite elements for general elliptic problem Manuscripta Geodaetica 1, 93-123 (1976) and Springer Lecture Notes m Math 564, 461-471 (1976) Zbl0353.65067 MR520343 · Zbl 0353.65067 [27] G STRANG, Approximation in the finite element method Num Math 19, (1972) Zbl0221.65174 MR305547 · Zbl 0221.65174 [28] M TAYLOR, Pseudodifferential Operators Princeton Univ Press, Princeton N J 1981 Zbl0453.47026 MR618463 · Zbl 0453.47026 [29] F TRÊVES, Pseudodifferential Operators Plenum Press New York, London 1980 MR597144 [30] W L WENDLAND, On applications and the convergence of boundary integral methods In Treatment of Integral Equations by Numerical Methods (ed T H Baker G F Miller), pp 463-476, Academic Press, London 1982 Zbl0561.65085 MR755378 · Zbl 0561.65085 [31] W L WENDLANDBoundary element methods and their asymptotic convergence In Theoretical Acoustics and Numerical Techniques (ed P Filippi), CISM Courses 277 Springer-Verlag, Wien, New York, 135 216 (1983) Zbl0618.65109 MR762829 · Zbl 0618.65109
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