25 research outputs found
High Order Cut Finite Element Methods for the Stokes Problem
We develop a high order cut finite element method for the Stokes problem
based on general inf-sup stable finite element spaces. We focus in particular
on composite meshes consisting of one mesh that overlaps another. The method is
based on a Nitsche formulation of the interface condition together with a
stabilization term. Starting from inf-sup stable spaces on the two meshes, we
prove that the resulting composite method is indeed inf-sup stable and as a
consequence optimal \emph{a~priori} error estimates hold
Full Gradient Stabilized Cut Finite Element Methods for Surface Partial Differential Equations
We propose and analyze a new stabilized cut finite element method for the
Laplace-Beltrami operator on a closed surface. The new stabilization term
provides control of the full gradient on the active mesh
consisting of the elements that intersect the surface. Compared to face
stabilization, based on controlling the jumps in the normal gradient across
faces between elements in the active mesh, the full gradient stabilization is
easier to implement and does not significantly increase the number of nonzero
elements in the mass and stiffness matrices. The full gradient stabilization
term may be combined with a variational formulation of the Laplace-Beltrami
operator based on tangential or full gradients and we present a simple and
unified analysis that covers both cases. The full gradient stabilization term
gives rise to a consistency error which, however, is of optimal order for
piecewise linear elements, and we obtain optimal order a priori error estimates
in the energy and norms as well as an optimal bound of the condition
number. Finally, we present detailed numerical examples where we in particular
study the sensitivity of the condition number and error on the stabilization
parameter.Comment: 20 pages, 4 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1507.0583
A CutFEM method for two-phase flow problems
In this article, we present a cut finite element method for two-phase
Navier-Stokes flows. The main feature of the method is the formulation of a
unified continuous interior penalty stabilisation approach for, on the one
hand, stabilising advection and the pressure-velocity coupling and, on the
other hand, stabilising the cut region. The accuracy of the algorithm is
enhanced by the development of extended fictitious domains to guarantee a well
defined velocity from previous time steps in the current geometry. Finally, the
robustness of the moving-interface algorithm is further improved by the
introduction of a curvature smoothing technique that reduces spurious
velocities. The algorithm is shown to perform remarkably well for low capillary
number flows, and is a first step towards flexible and robust CutFEM algorithms
for the simulation of microfluidic devices
A Nitsche-based cut finite element method for a fluid--structure interaction problem
We present a new composite mesh finite element method for fluid--structure
interaction problems. The method is based on surrounding the structure by a
boundary-fitted fluid mesh which is embedded into a fixed background fluid
mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The
coupling between the embedded and background fluid meshes is enforced using a
stabilized Nitsche formulation which allows us to establish stability and
optimal order \emph{a priori} error estimates,
see~\cite{MassingLarsonLoggEtAl2013}. We consider here a steady state
fluid--structure interaction problem where a hyperelastic structure interacts
with a viscous fluid modeled by the Stokes equations. We evaluate an iterative
solution procedure based on splitting and present three-dimensional numerical
examples.Comment: Revised version, 18 pages, 7 figures. Accepted for publication in
CAMCo