Monotone Multigrid Methods Based on Parametric Finite Elements

In this paper, a particular technique for the application of elementary multilevel ideas to problems with warped boundaries is studied in the context of the numerical simulation of elastic contact problems. Combining a general multilevel setting with a different perspective, namely an advanced geometric modeling point of view, we present a (monotone) multigrid method based on a hierarchy of parametric finite element spaces. For the construction, a full-dimensional parameterization of high order is employed which accurately represents the computational domain.The purpose of the volume parametric finite element discretization put forward here is two-fold. On the one hand, it allows for an elegant multilevel hierarchy to be used in preconditioners. On the other hand, it comes with particular advantages for the modeling of contact problems. After all, the long-term objective lies in an increased flexibility of h p-adaptive methods for contact problems.

[1]  Miloš Zlámal,et al.  The finite element method in domains with curved boundaries , 1973 .

[2]  W. J. Gordon,et al.  Transfinite element methods: Blending-function interpolation over arbitrary curved element domains , 1973 .

[3]  Helmut Harbrecht,et al.  A finite element method for elliptic problems with stochastic input data , 2010 .

[4]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[5]  Carsten Carstensen,et al.  P2Q2Iso2D = 2D isoparametric FEM in Matlab , 2006 .

[6]  Maharavo Randrianarivony,et al.  From Computer Aided Design to wavelet BEM , 2009, Comput. Vis. Sci..

[7]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[8]  R. Krause.  On the Multiscale Solution of Constrained Minimization Problems , 2008 .

[9]  Barbara Wohlmuth,et al.  Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D , 2007 .

[10]  Robert M. O'Bara,et al.  p-Version Mesh Generation Issues , 2002, IMR.

[11]  Padmanabhan Seshaiyer,et al.  Uniform hp convergence results for the mortar finite element method , 2000, Math. Comput..

[12]  C. Micchelli,et al.  Blossoming begets B -spline bases built better by B -patches , 1992 .

[13]  R. Kornhuber.  Adaptive monotone multigrid methods for nonlinear variational problems , 1997 .

[14]  Rolf Krause,et al.  Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization , 2009 .

[15]  Ernst P. Stephan,et al.  A priori error estimates for hp penalty BEM for contact problems in elasticity , 2007 .