65 research outputs found
An Eulerian Finite Element Method for PDEs in time-dependent domains
The paper introduces a new finite element numerical method for the solution
of partial differential equations on evolving domains. The approach uses a
completely Eulerian description of the domain motion. The physical domain is
embedded in a triangulated computational domain and can overlap the
time-independent background mesh in an arbitrary way. The numerical method is
based on finite difference discretizations of time derivatives and a standard
geometrically unfitted finite element method with an additional stabilization
term in the spatial domain. The performance and analysis of the method rely on
the fundamental extension result in Sobolev spaces for functions defined on
bounded domains. This paper includes a complete stability and error analysis,
which accounts for discretization errors resulting from finite difference and
finite element approximations as well as for geometric errors coming from a
possible approximate recovery of the physical domain. Several numerical
examples illustrate the theory and demonstrate the practical efficiency of the
method.Comment: 27 pages, 3 figures, 8 table
Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces
We present a new high order finite element method for the discretization of
partial differential equations on stationary smooth surfaces which are
implicitly described as the zero level of a level set function. The
discretization is based on a trace finite element technique. The higher
discretization accuracy is obtained by using an isoparametric mapping of the
volume mesh, based on the level set function, as introduced in [C. Lehrenfeld,
\emph{High order unfitted finite element methods on level set domains using
isoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting
trace finite element method is easy to implement. We present an error analysis
of this method and derive optimal order -norm error bounds. A
second topic of this paper is a unified analysis of several stabilization
methods for trace finite element methods. Only a stabilization method which is
based on adding an anisotropic diffusion in the volume mesh is able to control
the condition number of the stiffness matrix also for the case of higher order
discretizations. Results of numerical experiments are included which confirm
the theoretical findings on optimal order discretization errors and uniformly
bounded condition numbers.Comment: 28 pages, 5 figures, 1 tabl
A note on the penalty parameter in Nitsche's method for unfitted boundary value problems
Nitsche's method is a popular approach to implement Dirichlet-type boundary
conditions in situations where a strong imposition is either inconvenient or
simply not feasible. The method is widely applied in the context of unfitted
finite element methods. From the classical (symmetric) Nitsche's method it is
well-known that the stabilization parameter in the method has to be chosen
sufficiently large to obtain unique solvability of discrete systems. In this
short note we discuss an often used strategy to set the stabilization parameter
and describe a possible problem that can arise from this. We show that in
specific situations error bounds can deteriorate and give examples of
computations where Nitsche's method yields large and even diverging
discretization errors
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