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A uniformly well-conditioned, unfitted Nitsche method for interface problems: PartI
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0002-4911-467X
(English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170Article in journal (Other academic) Submitted
Abstract [en]

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented.The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and a variant of Nitsche's method enforces the jump conditions at the interface.In this method, the solutions on each side of the interface are extended to the entire domain, which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. Numerical experiments are presented showing optimal convergence order in the energy and $L^2$ norms, and also for pointwise errors. The presented results also show that the condition number of the system matrix is independent of the position of the interface relative to the grid.

National Category
URN: urn:nbn:se:kth:diva-33103OAI: diva2:413346
QS 20120328Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2012-03-28Bibliographically approved
In thesis
1. Numerical Methods for Fluid Interface Problems
Open this publication in new window or tab >>Numerical Methods for Fluid Interface Problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed.

In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure.

An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods.

A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011
Trita-CSC-A, ISSN 1653-5723 ; 2011:07
National Category
Computational Mathematics
urn:nbn:se:kth:diva-33111 (URN)978-91-7415-969-1 (ISBN)
Public defence
2011-05-20, Sal D3, Lindstedtsvägen 5, KTH, Stockholm, 14:21 (English)
Swedish e‐Science Research Center
QC 20110503Available from: 2011-05-03 Created: 2011-04-28 Last updated: 2012-05-24Bibliographically approved

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