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We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation...
We consider general surface energies, which are
weighted integrals over a closed surface with a weight function
depending on the position, the unit normal and
the mean curvature of the surface. Energies
of this form have applications in many areas, such as materials science,
biology and image processing. Often one is interested in finding
a surface that minimizes such an energy, which entails finding its first
variation with respect to perturbations of the surface.
We present a concise derivation...
Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on . Using an energy argument combined with a topological continuation argument and a spectral...
We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution...
We address the numerical approximation of the two-phase Stefan problem and discuss an adaptive finite element method based on rigorous a posteriori error estimation and refinement/coarsening. We also investigate how to restrict coarsening for the resulting method to be stable and convergent. We review implementation issues associated with bisection and conclude with simulations of a persistent corner singularity, for which adaptivity is an essential tool.
The phase relaxation model is a diffuse interface model with
small parameter which
consists of a parabolic PDE for temperature
and an ODE with double obstacles
for phase variable .
To decouple the system a semi-explicit Euler method with variable
step-size is used for time discretization, which requires
the stability constraint . Conforming piecewise
linear finite elements over highly graded simplicial meshes
with parameter are further employed for space discretization.
error
estimates...
We investigate the evolution of an almost flat membrane
driven by competition of the homogeneous, Frank, and
bending energies as well
as the coupling of the local order of the constituent molecules
of the membrane to its curvature.
We propose an alternative to
the model in [J.B. Fournier and P. Galatoa,
(1997) 1509–1520; N. Uchida,
(2002) 040902] which replaces
a Ginzburg-Landau penalization for the length of the
order parameter by a rigid constraint.
We introduce...
Phase-field models, the simplest of which is Allen–Cahn's
problem, are characterized by a small parameter that dictates
the interface thickness. These models naturally call for mesh adaptation
techniques, which rely on error control.
However, their error analysis usually deals with the
underlying non-monotone nonlinearity via a Gronwall argument which
leads to an exponential dependence on ε. Using an energy argument
combined with a
topological continuation argument and a spectral estimate,...
Motivated by the pricing of American options for baskets we
consider a parabolic variational inequality in a bounded
polyhedral domain with a continuous piecewise
smooth obstacle. We formulate a fully discrete method by using
piecewise linear finite elements in space and the backward Euler
method in time. We define an error estimator and show
that it gives an upper bound for the error in
(Ω)). The error estimator is localized in the
sense that the size of the elliptic residual is only relevant...
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