Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The $ K$-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates


Author: Thanh Tran
Journal: Math. Comp. 64 (1995), 501-513
MSC: Primary 65R20; Secondary 41A15, 65N38
DOI: https://doi.org/10.1090/S0025-5718-1995-1284671-2
MathSciNet review: 1284671
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Superconvergence in the $ {L^2}$-norm for the Galerkin approximation of the integral equation $ Lu = f$ is studied, where L is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let $ {u_h}$ be the Galerkin approximation to u. By using the K-operator, an operator that averages the values of $ {u_h}$, we will construct a better approximation than $ {u_h}$ itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm's equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.


References [Enhancements On Off] (What's this?)

  • [1] J.H. Bramble and A.H. Schatz, Higher order local accuracy by averaging in the finite element method, Mathematical Aspects of Finite Elements in Partial Differential Equations (Proc. Sympos., Math. Res. Center, Univ. of Wisconsin, Madison 1974) (Carl de Boor, ed.), Academic Press, New York, 1974, pp. 1-14. MR 0657964 (58:31903)
  • [2] -, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), 94-111. MR 0431744 (55:4739)
  • [3] G.A. Chandler, Superconvergence for second kind integral equations, The Application and Numerical Solutions of Integral Equations (R.S. Anderssen et al., eds.), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 103-117. MR 582986 (81h:45027)
  • [4] M. Costabel and E.P. Stephan, Duality estimates for the numerical solution of integral equations, Numer. Math. 54 (1988), 339-353. MR 971707 (90b:65245)
  • [5] G.C. Hsiao and W.L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), 299-315. MR 634453 (83j:45019)
  • [6] M.A. Jaswon and G.T. Symm, Integral equation methods in potential theory and elastostatics, Academic Press, New York, 1977. MR 0499236 (58:17147)
  • [7] R. Kress, Linear integral equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1979. MR 1007594 (90j:45001)
  • [8] R. Kress and I.H. Sloan, On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation, Numer. Math. 66 (1993), 199-214. MR 1245011 (95d:65108)
  • [9] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, Springer-Verlag, Berlin and New York, 1972.
  • [10] T. von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder--Singularitäten und Approximation mit Randelementmethoden, Dissertation, Technische Hochschule Darmstadt, 1989.
  • [11] J. Saranen, Local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on curves, Math. Comp. 48 (1987), 485-502. MR 878686 (88m:65208)
  • [12] I.H. Sloan, Error analysis of boundary integral methods, Acta Numer. 1 (1992), 287-339. MR 1165728 (94f:65117)
  • [13] E.P. Stephan and W.L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Anal. 18 (1984), 183-219. MR 767500 (86c:65141)
  • [14] V. Thomée, Higher order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), 652-660.
  • [15] T. Tran, The K-operator and the qualocation method for strongly elliptic equations on smooth curves, J. Integral Equations Appl. 5 (1993), 405-428. MR 1248498 (95a:65218)
  • [16] -, Local error estimates for the Galerkin method applied to strongly elliptic integral equations on open curves, Applied Mathematics Preprint AM93/13, University of New South Wales (June 1993), (to appear in SIAM J. Numer. Anal.). MR 1403555 (97g:65254)
  • [17] L.B. Wahlbin, Local behavior in finite element methods, Handbook of Numerical Analysis, Vol.11, Finite Element Methods (Part 1) (P.G. Ciarlet and J.L. Lions, eds.), Elsevier Science Publishers B.V., North-Holland, 1991, pp. 355-522. MR 1115238
  • [18] W.L. Wendland, Boundary element methods for elliptic problems, Mathematical Theory of Finite and Boundary Element Methods (A.H.Schatz, V.Thomée, and W.L.Wendland, eds.), Birkhäuser, Basel, 1990, pp. 219-276.
  • [19] Y. Yan and I.H. Sloan, Mesh grading for integral equations of the first kind with logarithmic kernel, SIAM J. Numer. Anal. 26 (1989), 574-587. MR 997657 (90f:65241)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 41A15, 65N38

Retrieve articles in all journals with MSC: 65R20, 41A15, 65N38


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1284671-2
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society