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

Tran, Thanh

doi:10.1090/S0025-5718-1995-1284671-2

The $K$-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates
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Math. Comp. 64 (1995), 501-513 Request permission

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

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. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 1–14. MR 0657964
J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), no. 137, 94–111. MR 431744, DOI 10.1090/S0025-5718-1977-0431744-9
G. A. Chandler, Superconvergence for second kind integral equations, 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. 103–117. MR 582986
Martin Costabel and Ernst P. Stephan, Duality estimates for the numerical solution of integral equations, Numer. Math. 54 (1988), no. 3, 339–353. MR 971707, DOI 10.1007/BF01396766
G. C. Hsiao and W. L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), no. 4, 299–315. MR 634453
M. A. Jaswon and G. T. Symm, Integral equation methods in potential theory and elastostatics, Computational Mathematics and Applications, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. MR 0499236
Rainer Kress, Linear integral equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989. MR 1007594, DOI 10.1007/978-3-642-97146-4
Rainer Kress and Ian H. Sloan, On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation, Numer. Math. 66 (1993), no. 2, 199–214. MR 1245011, DOI 10.1007/BF01385694

Non-homogeneous boundary value problems and applications

Randwertprobleme der Elastizitätstheorie für Polyeder—Singularitäten und Approximation mit Randelementmethoden

Jukka Saranen, Local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on curves, Math. Comp. 48 (1987), no. 178, 485–502. MR 878686, DOI 10.1090/S0025-5718-1987-0878686-9
Ian H. Sloan, Error analysis of boundary integral methods, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 287–339. MR 1165728, DOI 10.1017/s0962492900002294
Ernst P. Stephan and Wolfgang L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Applicable Anal. 18 (1984), no. 3, 183–219. MR 767500, DOI 10.1080/00036818408839520

Higher order local approximations to derivatives in the finite element method

31

Thanh Tran, The $K$-operator and the qualocation method for strongly elliptic equations on smooth curves, J. Integral Equations Appl. 5 (1993), no. 3, 405–428. MR 1248498, DOI 10.1216/jiea/1181075762
Thanh Tran, Local error estimates for the Galerkin method applied to strongly elliptic integral equations on open curves, SIAM J. Numer. Anal. 33 (1996), no. 4, 1484–1493. MR 1403555, DOI 10.1137/S0036142993251401
Lars B. Wahlbin, Local behavior in finite element methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR 1115238

Boundary element methods for elliptic problems

Y. Yan and I. H. Sloan, Mesh grading for integral equations of the first kind with logarithmic kernel, SIAM J. Numer. Anal. 26 (1989), no. 3, 574–587. MR 997657, DOI 10.1137/0726034