ALBERT---Software for scientific computations and applications.
A posteriori error control for the Allen–Cahn problem : circumventing Gronwall’s inequality
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...
Error control and adaptivity for a phase relaxation model
Adapting meshes and time-steps for phase change problems
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.
Error Control and Andaptivity for a Phase Relaxation Model
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...
error control for the Allen–Cahn problem: circumventing Gronwall's inequality
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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