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
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
- 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
- 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
- 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
- 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
- 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, DOI https://doi.org/10.1137/0520006
- 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, DOI https://doi.org/10.1051/proc%3A1999044
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- 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
- 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
- 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, DOI https://doi.org/10.1016/0022-1236%2876%2990035-5
- 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. 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
- 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
- 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
- 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
- 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
- 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, DOI https://doi.org/10.1002/cpa.20178
- 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
- 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
- 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, DOI https://doi.org/10.1002/mana.19901490106
References
- 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 (95j:35058)
- 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 (86m:65001b)
- Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Reprint of the 1978 original (North-Holland, Amsterdam); MR 0520174 (58:25001)
- 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 (91a:35078)
- ---, 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)
- 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 (53:8649)
- V. A. Kondrat$’$ev, Boundary value problems for elliptic equations in domains with conical or angular points, Transactions of the Moscow Mathematical Society for the year 1967, volume 16, American Mathematical Society, Providence, R.I., 1968, pp. 227–313. MR 0226187 (37:1777)
- V. A. Kondrat$’$ev and O. A. Oleĭnik, Boundary value problems for partial differential equations in nonsmooth domains, Russian Mathematical Surveys 38 (1983), 1–86. MR 695471 (85j:35002)
- 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)
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
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.
