394 research outputs found
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Geometric partial differential equations: Theory, numerics and applications
This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications
Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies
Published versio
Segmentation of Three-dimensional Images with Parametric Active Surfaces and Topology Changes
In this paper, we introduce a novel parametric method for segmentation of
three-dimensional images. We consider a piecewise constant version of the
Mumford-Shah and the Chan-Vese functionals and perform a region-based
segmentation of 3D image data. An evolution law is derived from energy
minimization problems which push the surfaces to the boundaries of 3D objects
in the image. We propose a parametric scheme which describes the evolution of
parametric surfaces. An efficient finite element scheme is proposed for a
numerical approximation of the evolution equations. Since standard parametric
methods cannot handle topology changes automatically, an efficient method is
presented to detect, identify and perform changes in the topology of the
surfaces. One main focus of this paper are the algorithmic details to handle
topology changes like splitting and merging of surfaces and change of the genus
of a surface. Different artificial images are studied to demonstrate the
ability to detect the different types of topology changes. Finally, the
parametric method is applied to segmentation of medical 3D images
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section
Geometric partial differential equations: Surface and bulk processes
The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems
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