Frontiers of Mathematics in China >
Partial expansion of a Lipschitz domain and some applications
Received date: 24 Jan 2011
Accepted date: 04 Jul 2011
Published date: 01 Apr 2012
Copyright
We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.
Jay Gopalakrishnan , Weifeng Qiu . Partial expansion of a Lipschitz domain and some applications[J]. Frontiers of Mathematics in China, 2012 , 7(2) : 249 -272 . DOI: 10.1007/s11464-012-0189-2
1 |
Amrouche C, Bernardi C, Dauge M, Girault V. Vector potentials in three-dimensional non-smooth domains. Math Methods Appl Sci, 1998, 21: 823-864
|
2 |
Arnold D N, Falk R S, Winther R. Preconditioning in H(div) and applications. Math Comp, 1997, 66: 957-984
|
3 |
Arnold D N, Falk R S, Winther R. Finite element exterior calculus: from Hodge theory to numerical stability. Bull Amer Math Soc (NS), 2010, 47: 281-353
|
4 |
Birman M, Solomyak M. Construction in a piecewise smooth domain of a function of the class H2 from the value of the conormal derivative. J Math Sov, 1990, 49: 1128-1136
|
5 |
Bonnet-Ben Dhia A-S, Hazard C, Lohrengel S. A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J Appl Math, 1999, 59: 2028-2044 (electronic)
|
6 |
Bramble J H. A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math Models Methods Appl Sci, 2003, 13: 361-371
|
7 |
Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, No 15. New York: Springer-Verlag, 1991
|
8 |
Buffa A, Costabel M, Sheen D. On traces for H(curl, Ω) in Lipschitz domains. J Math Anal Appl, 2002, 276: 845-867
|
9 |
Christiansen S H, Winther R. Smoothed projections in finite element exterior calculus. Math Comp, 2008, 77: 813-829
|
10 |
Cĺement Ph. Approximation by finite element functions using local regularization. RAIRO, Analyse Numérique, 1975, R-2 (9e année): 77-84
|
11 |
Demkowicz L, Gopalakrishnan J, Schöberl J. Polynomial extension operators. Part II. SIAM J Numer Anal, 2009, 47: 3293-3324
|
12 |
Demkowicz L, Gopalakrishnan J, Schöberl J. Polynomial extension operators. Part III. Math Comp, 2011,
|
13 |
Falk R S. Finite element methods for linear elasticity. In: Mixed Finite Elements, Compatibility Conditions, and Applications. Lectures given at the C.I.M.E. Summer School Held in Cetraro, Italy, June 26-July 1, 2006. Lecture Notes in Mathematics, Vol 1939. Berlin: Springer-Verlag, 2008, 160-194
|
14 |
Girault V, Raviart P-A. Finite ElementMethods for Navier-Stokes Equations. Springer Series in Computational Mathematics, No 5. New York: Springer-Verlag, 1986
|
15 |
Gopalakrishnan J, Oh M. Commuting smoothed projectors in weighted norms with an application to axisymmetric Maxwell equations. J Sci Comput, 2011,
|
16 |
Gopalakrishnan J, Pasciak J E. Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations. Math Comp, 2003, 72: 1-15 (electronic)
|
17 |
Grisvard P. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, No 24. Marshfield: Pitman Advanced Publishing Program, 1985
|
18 |
Hestenes M R. Extension of the range of a differentiable function. Duke Math J, 1941, 8: 183-192
|
19 |
Hiptmair R, Li J, Zhou J. Universal extension for Sobolev spaces of differential forms and applications. Tech Rep 2009-22, Eidgenössische Technische Hochschule, 2009
|
20 |
Hiptmair R, Toselli A. Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions. In: Parallel solution of partial differential equations (Minneapolis, MN, 1997). IMA VolMath Appl, Vol 120. New York: Springer, 2000, 181-208
|
21 |
Hofmann S, Mitrea M, Taylor M. Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains. J Geom Anal, 2007, 17: 593-647
|
22 |
Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach Science Publishers, 1963
|
23 |
McLean W. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge: Cambridge University Press, 2000
|
24 |
Něcas J. Les m´ethodes directes en th´eorie des ´equations elliptiques. Paris: Masson et Cie, ´Editeurs, 1967
|
25 |
Pasciak J E, Zhao J. Overlapping Schwarz methods in H(curl) on polyhedral domains. J Numer Math, 2002, 10: 221-234
|
26 |
Sch¨oberl J. Commuting quasi-interpolation operators for mixed finite elements. Tech Rep. ISC-01-10-MATH, Institute for Scientific Computation, Texas A&M University, College Station, 2001
|
27 |
Schöberl J. A multilevel decomposition result in H(curl). In: Wesseling P, Oosterlee C, Hemker P, eds. Proceedings of the 8th European Multigrid Conference, EMG 2005, TU Delft. 2008
|
28 |
Schöberl J. A posteriori error estimates for Maxwell equations. Math Comp, 2008, 77: 633-649
|
29 |
Scott L R, Vogelius M. Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél Math Anal Numér, 1985, 19: 111-143
|
30 |
Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No 30. Princeton: Princeton University Press, 1970
|
31 |
Vassilevski P S, Wang J P. Multilevel iterative methods for mixed finite element discretizations of elliptic problems. Numer Math, 1992, 63: 503-520
|
/
〈 | 〉 |