Partial expansion of a Lipschitz domain and some applications

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.

[1]  Michael E. Taylor,et al.  Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains , 2007 .

[2]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[3]  P. Vassilevski,et al.  Multilevel iterative methods for mixed finite element discretizations of elliptic problems , 1992 .

[4]  Douglas N. Arnold,et al.  Preconditioning in Hdiv and applications , 1997, Math. Comput..

[5]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[6]  Joseph E. Pasciak,et al.  Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations , 2003, Math. Comput..

[7]  Joachim Schöberl,et al.  Polynomial Extension Operators. Part II , 2009, SIAM J. Numer. Anal..

[8]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[9]  James H. Bramble.  A PROOF OF THE INF–SUP CONDITION FOR THE STOKES EQUATIONS ON LIPSCHITZ DOMAINS , 2003 .

[10]  L. R. Scott,et al.  Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials , 1985 .

[11]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[12]  Jun Zhao,et al.  Overlapping Schwarz methods in H(curl) on polyhedral domains , 2002, J. Num. Math..

[13]  M. Sh. Birman,et al.  Construction in a piecewise smooth domain of a function of the class H2 from the value of the conormal derivative , 1990 .

[14]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[15]  Jay Gopalakrishnan,et al.  Commuting Smoothed Projectors in Weighted Norms with an Application to Axisymmetric Maxwell Equations , 2011, Journal of Scientific Computing.

[16]  Joachim Sch Oberl.  COMMUTING QUASI INTERPOLATION OPERATORS FOR MIXED FINITE ELEMENTS , 2004 .

[17]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[18]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[19]  Ralf Hiptmair,et al.  Universal extension for Sobolev spaces of differential forms and applications , 2012 .

[20]  Snorre H. Christiansen,et al.  Smoothed projections in finite element exterior calculus , 2007, Math. Comput..

[21]  R. S. Falk.  Finite Element Methods for Linear Elasticity , 2008 .

[22]  R. S. Falk,et al.  PRECONDITIONING IN H (div) AND APPLICATIONS , 1997 .

[23]  M. Hestenes,et al.  Extension of the range of a differentiable function , 1941 .

[24]  Christophe Hazard,et al.  A Singular Field Method for the Solution of Maxwell's Equations in Polyhedral Domains , 1999, SIAM J. Appl. Math..

[25]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[26]  A. Buffa,et al.  On traces for H(curl,Ω) in Lipschitz domains , 2002 .

[27]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[28]  Joachim Schöberl,et al.  Polynomial extension operators. Part III , 2012, Math. Comput..

[29]  Andrea Toselli,et al.  Overlapping and Multilevel Schwarz Methods for Vector Valued Elliptic Problems in Three Dimensions , 2000 .

[30]  Joachim Schöberl,et al.  A posteriori error estimates for Maxwell equations , 2007, Math. Comput..