Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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

Abstract | References | Similar Articles | 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 [Enhancements On Off] (What's this?)

  • 1. A. K. Aziz and R. B. Kellogg, On homeomorphisms for an elliptic equation in domains with corners, Differential Integral Equations 8 (1995), no. 2, 333–352. MR 1296128
  • 2. Graham F. Carey and J. Tinsley Oden, Finite elements. Vol. II, The Texas Finite Element Series, II, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. A second course. MR 767804
  • 3. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • 4. 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
  • 5. Monique Dauge, 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, https://doi.org/10.1137/0520006
  • 6. Monique Dauge, 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 (English, with English and French summaries). MR 1689440, https://doi.org/10.1051/proc:1999044
  • 7. Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • 8. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • 9. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • 10. 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
  • 11. R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Analysis 21 (1976), no. 4, 397–431. MR 0404849
  • 12. 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
  • 13. V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for partial differential equations in nonsmooth domains, Uspekhi Mat. Nauk 38 (1983), no. 2(230), 3–76 (Russian). MR 695471
  • 14. 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
  • 15. V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991
  • 16. Alois Kufner, Weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian summaries. MR 664599
  • 17. 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
  • 18. 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.
  • 19. 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, https://doi.org/10.1002/cpa.20178
  • 20. 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
  • 21. 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
  • 22. Rouben Rostamian and Ana Maria Soane, Incompressible Fluid Flow and $ {C}^1$ Finite Elements, Book, in preparation, 2009.
  • 23. Ana Maria Soane, Variational problems in weighted Sobolev spaces with applications to Computational Fluid Dynamics, Ph.D. thesis, University of Maryland, Baltimore County, 2008.
  • 24. 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.
  • 25. 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, https://doi.org/10.1002/mana.19901490106

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35J20, 35Q30, 46E35, 76D05, 65N30

Retrieve articles in all journals with MSC (2000): 35J20, 35Q30, 46E35, 76D05, 65N30


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

DOI: https://doi.org/10.1090/S0033-569X-2010-01212-7
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.

American Mathematical Society