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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 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...
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,...
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