465 research outputs found
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
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
A trace finite element method for a class of coupled bulk-interface transport problems
In this paper we study a system of advection-diffusion equations in a bulk
domain coupled to an advection-diffusion equation on an embedded surface. Such
systems of coupled partial differential equations arise in, for example, the
modeling of transport and diffusion of surfactants in two-phase flows. The
model considered here accounts for adsorption-desorption of the surfactants at
a sharp interface between two fluids and their transport and diffusion in both
fluid phases and along the interface. The paper gives a well-posedness analysis
for the system of bulk-surface equations and introduces a finite element method
for its numerical solution. The finite element method is unfitted, i.e., the
mesh is not aligned to the interface. The method is based on taking traces of a
standard finite element space both on the bulk domains and the embedded
surface. The numerical approach allows an implicit definition of the surface as
the zero level of a level-set function. Optimal order error estimates are
proved for the finite element method both in the bulk-surface energy norm and
the -norm. The analysis is not restricted to linear finite elements and a
piecewise planar reconstruction of the surface, but also covers the
discretization with higher order elements and a higher order surface
reconstruction
Approximation of the determinant of large sparse symmetric positive definite matrices
This paper is concerned with the problem of approximating the determinant of A for a large sparse symmetric positive definite matrix A. It is shown that an efficient solution of this problem is obtained by using a sparse approximate inverse of A. The method is explained and theoretical properties are discussed. A posteriori error estimation techniques are presented. Furthermore, results of numerical experiments are given which illustrate the performance of this new method
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
The Loss Rank Principle for Model Selection
We introduce a new principle for model selection in regression and
classification. Many regression models are controlled by some smoothness or
flexibility or complexity parameter c, e.g. the number of neighbors to be
averaged over in k nearest neighbor (kNN) regression or the polynomial degree
in regression with polynomials. Let f_D^c be the (best) regressor of complexity
c on data D. A more flexible regressor can fit more data D' well than a more
rigid one. If something (here small loss) is easy to achieve it's typically
worth less. We define the loss rank of f_D^c as the number of other
(fictitious) data D' that are fitted better by f_D'^c than D is fitted by
f_D^c. We suggest selecting the model complexity c that has minimal loss rank
(LoRP). Unlike most penalized maximum likelihood variants (AIC,BIC,MDL), LoRP
only depends on the regression function and loss function. It works without a
stochastic noise model, and is directly applicable to any non-parametric
regressor, like kNN. In this paper we formalize, discuss, and motivate LoRP,
study it for specific regression problems, in particular linear ones, and
compare it to other model selection schemes.Comment: 16 page
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
An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces
The paper introduces a geometrically unfitted finite element method for the
numerical solution of the tangential Navier--Stokes equations posed on a
passively evolving smooth closed surface embedded in . The
discrete formulation employs finite difference and finite elements methods to
handle evolution in time and variation in space, respectively. A complete
numerical analysis of the method is presented, including stability, optimal
order convergence, and quantification of the geometric errors. Results of
numerical experiments are also provided
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