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Regularity estimates for elliptic boundary value problems in Besov spaces

Authors: Constantin Bacuta, James H. Bramble and Jinchao Xu
Journal: Math. Comp. 72 (2003), 1577-1595
MSC (2000): Primary 65N30, 46B70, 35J67, 35J05
Published electronically: December 18, 2002
MathSciNet review: 1986794
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Abstract: We consider the Dirichlet problem for Poisson's equation on a nonconvex plane polygonal domain $\Omega$. New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems. Some new sharp finite element error estimates are deduced.

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  • 1. R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975. MR 56:9247
  • 2. C. Bacuta, J. H. Bramble, J. Pasciak. New interpolation results and applications to finite element methods for elliptic boundary value problems. East-West J. Numer. Math. 9:179-198, 2001. MR 2002h:41053
  • 3. C. Bennett and R. Sharpley. Interpolation of Operators. Academic Press, New-York, 1988. MR 89e:46001
  • 4. J. Bergh, J. L$\ddot o$str$\ddot o$m. Interpolation Spaces. An Introduction, Springer-Verlag, New York, 1976. MR 58:2349
  • 5. D. Braess. Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, 1997. MR 98f:65002
  • 6. J. H. Bramble. Interpolation between Sobolev spaces in Lipschitz domains with an application to multigrid theory. Math. Comp., 64:1359-1365, 1995. MR 95m:46042
  • 7. J. H. Bramble and S. R. Hilbert. Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal., 7:113-124, 1970. MR 41:7819
  • 8. J. H. Bramble, J. Pasciak and P. S. Vassilevski. Computational scales of Sobolev norms with applications to preconditioning. Math. Comp., 69:463-480, 2000. MR 2000k:65088
  • 9. J. H. Bramble, J. Pasciak and J.Xu. Parallel multilevel preconditioners. Math. Comp., 55:1-22, 1990. MR 90k:65170
  • 10. J. H. Bramble and X. Zhang. The analysis of multigrid methods, in: Handbook for Numerical Analysis, Vol. VII, 173-415, P. Ciarlet and J.L. Lions, eds., North Holland, Amsterdam, 2000. MR 2001m:65183
  • 11. S. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994. MR 95f:65001
  • 12. P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 58:25001
  • 13. M. Dauge. Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341. Springer-Verlag, Berlin, 1988. MR 91a:35078
  • 14. M. Dobrowolski. Numerical Approximation of Elliptic Interface and Corner Problems. Rheinischen Friedrich-Wilhelms-Universität, Bonn,1981.
  • 15. P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. MR 86m:35044
  • 16. P. Grisvard. Singularities in Boundary Value Problems. Masson, Paris, 1992. MR 93h:35004
  • 17. P. Grisvard. Caracterisation de quelques espaces d'interpolation. Arc. Rat. Mech. Anal. 25:40-63, 1967. MR 35:4718
  • 18. R. B. Kellogg. Interpolation between subspaces of a Hilbert space, Technical note BN-719. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971.
  • 19. V. Kondratiev. Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc., 16:227-313, 1967. MR 37:1777
  • 20. V. A. Kozlov, V. G. Mazya and J. Rossmann. Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Mathematical Surveys and Monographs, vol. 52, 1997. MR 98f:35038
  • 21. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York, 1972. MR 50:2670
  • 22. J. L. Lions and P. Peetre. Sur une classe d'espaces d'interpolation. Institut des Hautes Etudes Scientifique. Publ.Math., 19:5-68, 1964. MR 29:2627
  • 23. S. A. Nazarov and B. A. Plamenevsky. Elliptic Problems in Domains with Piecewise Smooth Boundaries. Expositions in Mathematics, vol. 13, de Gruyter, New York, 1994. MR 95h:35001
  • 24. J. Necas. Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967. MR 37:3168
  • 25. P. Oswald. Multilevel Finite Element Approximation. B. G. Teubner, Stuttgart, 1994. MR 95k:65110
  • 26. L. Wahlbin. On the sharpness of certain local estimates for $H^1_0$projections into finite element spaces: In fluence of a reentrant corner. Math. Comp., 42:1-8, 1984. MR 86b:65129
  • 27. J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Review, 34:581-613, December 1992. MR 93k:65029
  • 28. K. Yosida. Lectures on Differential and Integral Equations. Dover Publications, Inc., New York, 1991. MR 92a:34002

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Additional Information

Constantin Bacuta
Affiliation: Dept. of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

James H. Bramble
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843

Jinchao Xu
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Keywords: Interpolation spaces, finite element method, multilevel decomposition, shift theorems, Besov spaces
Received by editor(s): January 26, 2002
Received by editor(s) in revised form: March 25, 2002
Published electronically: December 18, 2002
Additional Notes: The work of the second author was supported in part under NSF Grant No. DMS-9973328.
The work of the third author was supported under NSF Grant No. DMS-0074299.
Article copyright: © Copyright 2002 American Mathematical Society

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