160 research outputs found
Error analysis of a space-time finite element method for solving PDEs on evolving surfaces
In this paper we present an error analysis of an Eulerian finite element
method for solving parabolic partial differential equations posed on evolving
hypersurfaces in , . The method employs discontinuous
piecewise linear in time -- continuous piecewise linear in space finite
elements and is based on a space-time weak formulation of a surface PDE
problem. Trial and test surface finite element spaces consist of traces of
standard volumetric elements on a space-time manifold resulting from the
evolution of a surface. We prove first order convergence in space and time of
the method in an energy norm and second order convergence in a weaker norm.
Furthermore, we derive regularity results for solutions of parabolic PDEs on an
evolving surface, which we need in a duality argument used in the proof of the
second order convergence estimate
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
Trace Finite Element Methods for PDEs on Surfaces
In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject
A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces
The paper studies a method for solving elliptic partial differential
equations posed on hypersurfaces in , . The method allows
a surface to be given implicitly as a zero level of a level set function. A
surface equation is extended to a narrow-band neighborhood of the surface. The
resulting extended equation is a non-degenerate PDE and it is solved on a bulk
mesh that is unaligned to the surface. An unfitted finite element method is
used to discretize extended equations. Error estimates are proved for finite
element solutions in the bulk domain and restricted to the surface. The
analysis admits finite elements of a higher order and gives sufficient
conditions for archiving the optimal convergence order in the energy norm.
Several numerical examples illustrate the properties of the method.Comment: arXiv admin note: text overlap with arXiv:1301.470
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
Two-level method and some a priori estimates in unsteady Navier-Stokes calculations
AbstractA two-level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible Navier-Stokes flows. The method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid, the scaling between these two grids is superlinear. Approximation, stability, and convergence aspects of a fully discrete scheme are considered. Stability properties of the two-level scheme are compared with those for a commonly used semi-implicit scheme, some new estimates are also proved for the latter
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