Remarks on Sobolev norms of fractional orders

When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on the whole domain? On what do the comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove some misconceptions by counter-examples

Multilevel methods for the h-, p-, and hp-versions of the boundary element method

AbstractIn this paper we give an overview on the definition of finite element spaces for the h-, p-, and hp-version of the BEM along with preconditioners of additive Schwarz type. We consider screen problems (with a hypersingular or a weakly singular integral equation of first kind on an open surface Γ) as model problems. For the hypersingular integral equation and the h-version with piecewise bilinear functions on a coarse and a fine grid we analyze a preconditioner by iterative substructuring based on a non-overlapping decomposition of Γ. We prove that the condition number of the preconditioned linear system behaves polylogarithmically in H/h. Here H is the size of the subdomains and h is the size of the elements. For the hp-version and the hypersingular integral equation we comment in detail on an additive Schwarz preconditioner which uses piecewise polynomials of high degree on the fine grid and yields also a polylogarithmically growing condition number. For the weakly singular integral equation, where no continuity of test and trial functions across the element boundaries has to been enforced, the method works for nonuniform degree distributions as well. Numerical results supporting our theory are reported

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

The article is split into four parts. First, we present the symmetric finite element/boundary element-coupling method. Second, we address the choices of appropriate preconditioners for the resulting discrete system when h- and p-versions are performed. Third, we discuss contact problems which are reduced to variational inequalities. Finally, we show the practical applicability of the finite element/boundary element-coupling method by applying it to a metal turning process. Here the viscoplastic work piece is modelled with finite elements and the linear elastic work tool (milling cutter) is modelled with boundary elements. This leads to an efficient and fast numerical method to simulate the metal turning process and to predict failure of the thermal shrink fit which holds the milling cutter. References A. Chernov, M. Maischak and E. P. Stephan. A priori error estimates for hp penalty bem for contact problems in elasticity. Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3871--3880. doi:10.1016/j.cma.2006.10.044 A. Chernov, M. Maischak and E. P. Stephan. hp-mortar boundary element methodfor two-body contact problems with friction. Math. Methods Appl. Sci., published online, doi:10.1002/mma.1005 A. Chernov and E. P. Stephan. Adaptive bem for contact problems with friction. IUTAM Symposium on Computational Methods in Contact Mechanics, 113--122, IUTAM Bookser., 3, Springer, Dordrecht, 2007. M. Costabel and E. P. Stephan. Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (1990), no. 5, 1212--1226. B. Denkena, E. P. Stephan, M. Maischak, D. Heinisch, M. Andres. Numerical computation methods for process-oriented structures in metal chipping. Proceedings of 1st International Conference on Process Machine Interaction (PMI). (2008), 247--258. B. Guo and E. P. Stephan. The hp-version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains. Numer. Math. 80 (1998), no. 1, 87--107. E. W. Hart. Constitutive relations for the nonelastic deformation of metals, Journal of Enginering Materials and Technology. 98 (1976), 193--202. N. Heuer, F. Leydecker and E. P. Stephan. An iterative substructuring method for the hp-version of the bem on quasi-uniform triangular meshes, Num.Meth.PDEs. 23 (2007), 879--903. doi:10.1002/num.20259 N. Heuer, M. Maischak and E. P. Stephan. Preconditioned minimum residual iteration for the hp-version of the coupled fem/bem with quasi-uniform meshes. Numer. Linear Algebra Appl. 6 (1999), no. 6, 435--456. N. Heuer and E. P. Stephan. An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in $R^3$, IMA J. Numer. Analysis. 21 (2001), 265--283. doi:10.1093/imanum/21.1.265 I. Babuska, A. Craig, J. Mandel and J. Pitkaeranta. Efficient preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal. 28 (1991), 624--661. N. Heuer, E. P. Stephan and T. Tran. Multilevel additive Schwarz method for the hp-version of the Galerkin boundary element method. Math.Comp.. 67 (1998), no. 222, 501--518. M. Maischak and E. P. Stephan. The hp-version of the boundary element method in $R^3$. The basic approximation results. Math. Meth. Appl. Sci. 20 (1997), 461--476. M. Maischak and E. P. Stephan. Adaptive hp-versions of boundary element methods for elastic contact problems, Comp.Mech. 39 (2007), 597--607. doi:10.1007/s00466-006-0109-y S. Mukherjee and A. Chandra. Boundary element formulations for large strain-large deformation problems of plasticity and viscoplasticity, Developments in Boundary Element Methods, 1984, editors P. K. Banerjee and S. Mukherjee, 27--58. P. Mund and E. P. Stephan. Adaptive coupling and fast solution of fem-bem equations for parabolic-elliptic interface problems. Math. Methods Appl. Sci. 20 (1997), no. 5, 403--423. P. Mund and E. P. Stephan. An adaptive two-level method for the coupling of nonlinear fem-bem equations. SIAM J. Numer. Anal. 36 (1999), no. 4, 1001--1021. P. Seshaiyer and M. Suri. Uniform hp convergence results for the mortar finite element method. Math. Comp. 69 (2000), no. 230, 521--546. E. P. Stephan. Coupling of finite elements and boundary elements for some nonlinear interface problems, Comput. Methods Appl. Mech. Engrg. 101 (1992), no. 1--3, 61--72. E. P. Stephan. The hp-boundary element method for solving $2$- and 3-dimensional problems. Comput. Methods Appl. Mech. Engrg. 133 (1996), no. 3--4, 183--208. E. P. Stephan. Coupling of Boundary Element Methods and Finite Element Methods, Encyclopedia of Computational Mechanics. Vol. 1, Chapter 13. 2004, John Wiley and Sons. ISBN: 0-470-84699-2. I. Babuska, B. Q. Guo and E. P. Stephan. The hp-version of the boundary element method with geometric mesh on polygonal domains. Computer Methods in Applied Mechanics and Engineering 80 (1990), 319--325. E. P. Stephan, S. Geyn, M. Maischak and M. Andres. A boundary element / finite element procedure for metal chipping. CMM-2007, June 19--22, 2007, \unhbox \voidb@x \setbox \z@ \hbox {L}\hbox to\wd \z@ {\hss \@xxxii L}odz--Spala, Poland. T. Tran and E. P. Stephan. Additive Schwarz methods for the h-version boundary element method. Applicable Analysis. 60 (1996), no. 1--2, 63--84. T. Tran and E. P. Stephan. Additive {Schwarz} algorithms for the p-version of the Galerkin boundary element method. Numer. Math.. 85 (2000), no. 3, 433--468. T. Tran and E. P. Stephan. An overlapping additive Schwarz preconditioner for boundary element approximations to the Laplace screen and Lame crack problems, J. Numer. Math. 12 (2004), 311--330. doi:10.1515/1569395042571265 P. Wriggers. Computational Contact Mechanics, 2002, John Wiley and Sons. E. Zeidler. Nonlinear Functional Analysis and its Applications. IV. Springer, New York, 1988. C. Carstensen and E. P. Stephan. Adaptive coupling of boundary elements and finite elements. RAIRO Modl. Math. Anal. Numer. 29 (1995), no. 7, 779--817. A. Chernov. Nonconforming boundary elements and finite elements for interface and contact problems with friction; hp-version for mortar, penalty and Nitsche's methods, PhD Thesis, Universitaet Hannover, 2007