289 research outputs found
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model
An implicit variable-step BDF2 scheme is established for solving the space
fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived
from a gradient flow in the negative order Sobolev space ,
. The Fourier pseudo-spectral method is applied for the spatial
approximation. The proposed scheme inherits the energy dissipation law in the
form of the modified discrete energy under the sufficient restriction of the
time-step ratios. The convergence of the fully discrete scheme is rigorously
provided utilizing the newly proved discrete embedding type convolution
inequality dealing with the fractional Laplacian. Besides, the mass
conservation and the unique solvability are also theoretically guaranteed.
Numerical experiments are carried out to show the accuracy and the energy
dissipation both for various interface widths. In particular, the
multiple-time-scale evolution of the solution is captured by an adaptive
time-stepping strategy in the short-to-long time simulation
Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
Time-fractional partial differential equations are nonlocal in time and show
an innate memory effect. In this work, we propose an augmented energy
functional which includes the history of the solution. Further, we prove the
equivalence of a time-fractional gradient flow problem to an integer-order one
based on our new energy. This equivalence guarantees the dissipating character
of the augmented energy. The state function of the integer-order gradient flow
acts on an extended domain similar to the Caffarelli-Silvestre extension for
the fractional Laplacian. Additionally, we apply a numerical scheme for solving
time-fractional gradient flows, which is based on kernel compressing methods.
We illustrate the behavior of the original and augmented energy in the case of
the Ginzburg-Landau energy functional
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