30 research outputs found
An Eulerian space-time finite element method for diffusion problems on evolving surfaces
In this paper, we study numerical methods for the solution of partial
differential equations on evolving surfaces. The evolving hypersurface in
defines a -dimensional space-time manifold in the space-time
continuum . We derive and analyze a variational formulation for
a class of diffusion problems on the space-time manifold. For this variational
formulation new well-posedness and stability results are derived. The analysis
is based on an inf-sup condition and involves some natural, but non-standard,
(anisotropic) function spaces. Based on this formulation a discrete in time
variational formulation is introduced that is very suitable as a starting point
for a discontinuous Galerkin (DG) space-time finite element discretization.
This DG space-time method is explained and results of numerical experiments are
presented that illustrate its properties.Comment: 22 pages, 5 figure
Cut Finite Elements for Convection in Fractured Domains
We develop a cut finite element method (CutFEM) for the convection problem in
a so called fractured domain which is a union of manifolds of different
dimensions such that a dimensional component always resides on the boundary
of a dimensional component. This type of domain can for instance be used
to model porous media with embedded fractures that may intersect. The
convection problem can be formulated in a compact form suitable for analysis
using natural abstract directional derivative and divergence operators. The cut
finite element method is based on using a fixed background mesh that covers the
domain and the manifolds are allowed to cut through a fixed background mesh in
an arbitrary way. We consider a simple method based on continuous piecewise
linear elements together with weak enforcement of the coupling conditions and
stabilization. We prove a priori error estimates and present illustrating
numerical examples
Numerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces
In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level
set based method for numerical solution of partial differential equations (PDEs) of
the reaction-diffusion-convection type on an evolving-in-time hypersurface Γ (t). In a
series of numerical experiments we study the accuracy and robustness of the proposed
scheme and demonstrate that the method is applicable to practical models
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 Trace Finite Element Method for Vector-Laplacians on Surfaces
We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D
domain, which results from the modeling of surface fluids based on exterior
Cartesian differential operators. The main topic of this paper is the
development and analysis of a finite element method for the discretization of
this surface partial differential equation. We apply the trace finite element
technique, in which finite element spaces on a background shape-regular
tetrahedral mesh that is surface-independent are used for discretization. In
order to satisfy the constraint that the solution vector field is tangential to
the surface we introduce a Lagrange multiplier. We show well-posedness of the
resulting saddle point formulation. A discrete variant of this formulation is
introduced which contains suitable stabilization terms and is based on trace
finite element spaces. For this method we derive optimal discretization error
bounds. Furthermore algebraic properties of the resulting discrete saddle point
problem are studied. In particular an optimal Schur complement preconditioner
is proposed. Results of a numerical experiment are included