17 research outputs found
Unfitted finite element methods using bulk meshes for surface partial differential equations
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ⊂Rn+1, is embedded in a polyhedral domain in Rn+1 consisting of a union, Th, of (n+1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on Th. Our first method is a sharp interface method, \emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes \emph{SIF} from the method of [42]. The second method, \emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width O(h). \emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L2 and H1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence
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 cut finite element method for coupled bulk-surface problems on time-dependent domains
In this contribution we present a new computational method for coupled
bulk-surface problems on time-dependent domains. The method is based on a
space-time formulation using discontinuous piecewise linear elements in time
and continuous piecewise linear elements in space on a fixed background mesh.
The domain is represented using a piecewise linear level set function on the
background mesh and a cut finite element method is used to discretize the bulk
and surface problems. In the cut finite element method the bilinear forms
associated with the weak formulation of the problem are directly evaluated on
the bulk domain and the surface defined by the level set, essentially using the
restrictions of the piecewise linear functions to the computational domain. In
addition a stabilization term is added to stabilize convection as well as the
resulting algebraic system that is solved in each time step. We show in
numerical examples that the resulting method is accurate and stable and results
in well conditioned algebraic systems independent of the position of the
interface relative to the background mesh
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
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