Partial expansion of a Lipschitz domain and some applications

doi:10.1007/s11464-012-0189-2

Front Math Chin

2012, Vol. 7

Issue (2) : 249-272 https://doi.org/10.1007/s11464-012-0189-2

RESEARCH ARTICLE

Partial expansion of a Lipschitz domain and some applications

Jay Gopalakrishnan¹, Weifeng Qiu²(

)

1. Department of Mathematics, Portland State University, Portland, OR 97207, USA; 2. Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, USA

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Abstract

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.

Keywords Lipschitz domain regular decomposition mixed boundary condition transversal vector field extension operator Schwarz preconditioner bounded cochain projector divergence curl Sch?berl projector

Corresponding Author(s): Qiu Weifeng,Email:qiuxa001@ima.umn.edu

Issue Date: 01 April 2012

Cite this article:

Jay Gopalakrishnan,Weifeng Qiu. Partial expansion of a Lipschitz domain and some applications[J]. Front Math Chin, 2012, 7(2): 249-272.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0189-2
https://academic.hep.com.cn/fmc/EN/Y2012/V7/I2/249

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 doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
2	Arnold D N, Falk R S, Winther R. Preconditioning in H(div) and applications. Math Comp , 1997, 66: 957-984 doi: 10.1090/S0025-5718-97-00826-0
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 doi: 10.1090/S0273-0979-10-01278-4
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 doi: 10.1007/BF02208708
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 doi: 10.1142/S0218202503002544
7	Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics , No 15. New York: Springer-Verlag, 1991 doi: 10.1007/978-1-4612-3172-1
8	Buffa A, Costabel M, Sheen D. On traces for H(curl, Ω) in Lipschitz domains. J Math Anal Appl , 2002, 276: 845-867 doi: 10.1016/S0022-247X(02)00455-9
9	Christiansen S H, Winther R. Smoothed projections in finite element exterior calculus. Math Comp , 2008, 77: 813-829 doi: 10.1090/S0025-5718-07-02081-9
10	C?ement Ph. Approximation by finite element functions using local regularization. RAIRO, Analyse Numérique , 1975, R-2 (9e année): 77-84
11	Demkowicz L, Gopalakrishnan J, Sch?berl J. Polynomial extension operators. Part II. SIAM J Numer Anal , 2009, 47: 3293-3324 doi: 10.1137/070698798
12	Demkowicz L, Gopalakrishnan J, Sch?berl J. Polynomial extension operators. Part III. Math Comp , 2011, doi: 10.1090/S0025-5718-2011-02536-6
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 doi: 10.1007/978-3-642-61623-5
15	Gopalakrishnan J, Oh M. Commuting smoothed projectors in weighted norms with an application to axisymmetric Maxwell equations. J Sci Comput , 2011, doi: 10.1007/s10915-011-9513-3
16	Gopalakrishnan J, Pasciak J E. Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations. Math Comp , 2003, 72: 1-15 (electronic) doi: 10.1090/S0025-5718-01-01406-5
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 doi: 10.1215/S0012-7094-41-00812-8
19	Hiptmair R, Li J, Zhou J. Universal extension for Sobolev spaces of differential forms and applications. Tech Rep 2009-22, Eidgen?ssische 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 doi: 10.1007/978-1-4612-1176-1_8
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 doi: 10.1007/BF02937431
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 doi: 10.1515/JNMA.2002.221
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	Sch?berl 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	Sch?berl 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 doi: 10.1007/BF01385872

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