2012-11-09

Nitsche-based finite element methods for the Stokes problem on cut and composite meshes – analysis, implementation and applications

Andre Massing, PhD, Simula Research Laboratory AS, Fornebu (Norway)

22 Nov 2012, 17:00; Location: S4|10-1

Multi-domain and multi-physics problems with moving interfaces and parameter studies with changing geometric domains can be severely limited by the use of conforming meshes when complex geometries in three spatial dimensions are involved. To overcome these limitations, several fixed-grid methods based on XFEM, Nitsche's method and related approaches have been investigated and shown promising results in recent years.

Nitsche's method is a general approach to formulate boundary and interface conditions in a weak sense.  It presents a promising alternative to existing Lagrange multiplier and penalty methods due to its analytical properties. Consequently, Nitsche's method has recently been employed to formulate new fictitious domain approaches and domain decomposition methods based on composite meshes and arbitrary cutting interfaces.

In the first part of this talk, we review how sophisticated algorithms and data structures from the field of computational geometry can be employed to efficiently implement schemes based on fictitious and overlapping domains. These techniques are part of an emerging framework for Nitsche-type methods based on the FEniCS project.  In the second part, we present some very recent results for a Nitsche-based formulation of the Stokes problem on both fictitious and overlapping domains.  Using so-called ghost-penalties, optimal a priori estimates are obtained. Moreover, the condition number of stiffness matrix can be bounded independently of the location of the interface and the domain boundary.  The talk concludes by demonstrating various application of the given formulations.


References

[1] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes, “A stabilized Nitsche fictitious domain method for the Stokes problem”, submitted for publication, 2012 (available as arXiv preprint arXiv:1206.1933)

[2] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes, “A stabilized Nitsche overlapping mesh method for the Stokes problem”, submitted for publication, 2012 (available as arXiv preprint arXiv:1205.6317)

[3] A. Massing, M. G. Larson, and A. Logg, “Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D ”, SIAM J. Sci. Comput., accepted, 2012 (available as arXiv preprint arXiv:1210.7076)

[4] A. Massing, M. G. Larson and A. Logg, "Towards an Implementation of Nitsche’s Method on Overlapping Meshes in 3D", AIP Conference Proceedings 1281, 2010

Category: CE Seminar

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