Variational problems in weighted Sobolev spaces on non-smooth domains
Authors:
Ana Maria Soane and Rouben Rostamian
Journal:
Quart. Appl. Math. 68 (2010), 439-458
MSC (2000):
Primary 35J20; Secondary 35Q30, 46E35, 76D05, 65N30
DOI:
https://doi.org/10.1090/S0033-569X-2010-01212-7
Published electronically:
May 19, 2010
MathSciNet review:
2676970
Full-text PDF Free Access
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Additional Information
Abstract:
We study the Poisson problem $-\Delta u = f$ and the Helmholtz problem $-\Delta u + \lambda u = f$ in bounded domains with angular corners in the plane and $u=0$ on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space $H^1$ but not in $H^2$ in general, even though $f$ may be very regular. We formulate these as variational problems in weighted Sobolev spaces and prove existence and uniqueness of solutions in what would be weighted counterparts of $H^2 \cap H^1_0$.
The specific forms of our variational formulations are motivated by, and are particularly suited to, applying a finite element scheme for solving the time-dependent Navier-Stokes equations of fluid mechanics.
References
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References
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- ---, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489 (90b:35191)
- ---, Singularities of corner problems and problems of corner singularities, Actes du 30ème Congrès d’Analyse Numérique: CANum ’98 (Arles, 1998), ESAIM Proc., vol. 6, Soc. Math. Appl. Indust., Paris, 1999, pp. 19–40 (electronic). MR 1689440 (2001g:35054)
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683 (86m:35044)
- ---, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris, 1992. MR 1173209 (93h:35004)
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- 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, R.I., 1997. MR 1469972 (98f:35038)
- ---, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, R.I., 2001. MR 1788991 (2001i:35069)
- Alois Kufner, Weighted Sobolev spaces, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. MR 664599 (84e:46029)
- Alois Kufner and Anna-Margarete Sändig, Some applications of weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 100, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987, With German, French and Russian summaries. MR 926688 (89h:35096)
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.
- Jian-Guo Liu, Jie Liu, and Robert L. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math. 60 (2007), no. 10, 1443–1487. MR 2342954 (2008k:76039)
- Serge Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), no. 3, 411–429. MR 1457079 (98k:35044)
- John E. Osborn, Regularity of solutions of the Stokes problem in a polygonal domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), Academic Press, New York, 1976, pp. 393–411. MR 0467032 (57:6901)
- Rouben Rostamian and Ana Maria Soane, Incompressible Fluid Flow and ${C}^1$ Finite Elements, Book, in preparation, 2009.
- Ana Maria Soane, Variational problems in weighted Sobolev spaces with applications to Computational Fluid Dynamics, Ph.D. thesis, University of Maryland, Baltimore County, 2008.
- Ana Maria Soane, Manul Suri, and Rouben Rostamian, The optimal convergence rate of a ${C}^1$ finite element method for non-smooth domains, Journal of Computational and Applied Mathematics 233 (2010), no. 10, 2711–2723.
- T. von Petersdorff and E. P. Stephan, Decompositions in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron, Math. Nachr. 149 (1990), 71–103. MR 1124795 (92h:35053)
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Additional Information
Ana Maria Soane
Affiliation:
Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore 21250, Maryland
Address at time of publication:
MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy
Email:
asoane@umbc.edu
Rouben Rostamian
Affiliation:
Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore 21250, Maryland
Email:
rostamian@umbc.edu
Keywords:
Poisson problem,
Helmholtz problem,
corner singularities,
weighted Sobolev spaces,
finite elements,
Navier-Stokes equations
Received by editor(s):
August 12, 2008
Published electronically:
May 19, 2010
Article copyright:
© Copyright 2010
Brown University
The copyright for this article reverts to public domain 28 years after publication.