Regularity estimates for elliptic boundary value problems in Besov spaces
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- by Constantin Bacuta, James H. Bramble and Jinchao Xu PDF
- Math. Comp. 72 (2003), 1577-1595 Request permission
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.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- C. Bacuta, J. H. Bramble, and J. E. Pasciak, New interpolation results and applications to finite element methods for elliptic boundary value problems, East-West J. Numer. Math. 9 (2001), no. 3, 179–198. MR 1862579
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Dietrich Braess, Finite elements, Cambridge University Press, Cambridge, 1997. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German original by Larry L. Schumaker. MR 1463151, DOI 10.1007/978-3-662-07233-2
- James H. Bramble, Interpolation between Sobolev spaces in Lipschitz domains with an application to multigrid theory, Math. Comp. 64 (1995), no. 212, 1359–1365. MR 1308447, DOI 10.1090/S0025-5718-1995-1308447-2
- J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
- James H. Bramble, Joseph E. Pasciak, and Panayot S. Vassilevski, Computational scales of Sobolev norms with application to preconditioning, Math. Comp. 69 (2000), no. 230, 463–480. MR 1651742, DOI 10.1090/S0025-5718-99-01106-0
- James H. Bramble, Joseph E. Pasciak, and Jinchao Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), no. 191, 1–22. MR 1023042, DOI 10.1090/S0025-5718-1990-1023042-6
- James H. Bramble and Xuejun Zhang, The analysis of multigrid methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173–415. MR 1804746
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
- M. Dobrowolski. Numerical Approximation of Elliptic Interface and Corner Problems. Rheinischen Friedrich-Wilhelms-Universität, Bonn,1981.
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
- P. Grisvard, Caractérisation de quelques espaces d’interpolation, Arch. Rational Mech. Anal. 25 (1967), 40–63 (French). MR 213864, DOI 10.1007/BF00281421
- 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.
- V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
- V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972, DOI 10.1090/surv/052
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
- J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343
- Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
- Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
- Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR 1312165, DOI 10.1007/978-3-322-91215-2
- Lars B. Wahlbin, On the sharpness of certain local estimates for $\r H{}^{1}$ projections into finite element spaces: influence of a re-entrant corner, Math. Comp. 42 (1984), no. 165, 1–8. MR 725981, DOI 10.1090/S0025-5718-1984-0725981-7
- Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
- K\B{o}saku Yosida, Lectures on differential and integral equations, Dover Publications, Inc., New York, 1991. Translated from the Japanese; Reprint of the 1960 translation. MR 1105769
Additional Information
- Constantin Bacuta
- Affiliation: Dept. of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
- Email: bacuta@math.psu.edu
- James H. Bramble
- Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
- Email: bramble@math.tamu.edu
- Jinchao Xu
- Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 228866
- Email: xu@math.psu.edu
- 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. - © Copyright 2002 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 1577-1595
- MSC (2000): Primary 65N30, 46B70, 35J67, 35J05
- DOI: https://doi.org/10.1090/S0025-5718-02-01502-8
- MathSciNet review: 1986794