A robust Nitsche’s formulation for interface problems

Abstract In this work, we propose a novel weighting for the interfacial consistency terms arising in a Nitsche variational form. We demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method. Consequently, we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings; ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of (a) elements with arbitrarily small volume fractions, (b) large material heterogeneities and (c) both large heterogeneities as well as arbitrarily small elements. We then highlight the accuracy and efficiency of the proposed formulation through numerical examples, focusing particular attention on interfacial quantities of interest.

[1]  Nicolas Moës,et al.  A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method , 2009 .

[2]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[3]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[4]  John E. Dolbow,et al.  Residual-free bubbles for embedded Dirichlet problems , 2008 .

[5]  Tod A. Laursen,et al.  Mortar contact formulations for deformable–deformable contact: Past contributions and new extensions for enriched and embedded interface formulations , 2012 .

[6]  Erik Burman,et al.  A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems , 2006, SIAM J. Numer. Anal..

[7]  Angelo Simone,et al.  Partition of unity-based discontinuous elements for interface phenomena: computational issues , 2004 .

[8]  P. Hansbo,et al.  A finite element method for the simulation of strong and weak discontinuities in solid mechanics , 2004 .

[9]  Shun Zhang,et al.  Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations , 2011, SIAM J. Numer. Anal..

[10]  J. Dolbow,et al.  Robust imposition of Dirichlet boundary conditions on embedded surfaces , 2012 .

[11]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[12]  Paolo Zunino,et al.  An unfitted interface penalty method for the numerical approximation of contrast problems , 2011 .

[13]  Paolo Zunino.  Discontinuous Galerkin Methods Based on Weighted Interior Penalties for Second Order PDEs with Non-smooth Coefficients , 2009, J. Sci. Comput..

[14]  Adrian J. Lew,et al.  Optimal convergence of a discontinuous-Galerkin-based immersed boundary method* , 2011 .

[15]  John E. Dolbow,et al.  Stable imposition of stiff constraints in explicit dynamics for embedded finite element methods , 2012 .

[16]  Tod A. Laursen,et al.  On methods for stabilizing constraints over enriched interfaces in elasticity , 2009 .

[17]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[18]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[19]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[20]  A. Ern,et al.  A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity , 2008 .

[21]  Tod A. Laursen,et al.  A Nitsche embedded mesh method , 2012 .

[22]  Michael Griebel,et al.  A Particle-Partition of Unity Method Part V: Boundary Conditions , 2003 .

[23]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[24]  Rolf Stenberg,et al.  Nitsche's method for general boundary conditions , 2009, Math. Comput..

[25]  John E. Dolbow,et al.  On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method , 2004 .

[26]  Isaac Harari,et al.  An efficient finite element method for embedded interface problems , 2009 .

[27]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[28]  Isaac Harari,et al.  A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces , 2007 .

[29]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[30]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .