90 research outputs found
Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows
This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem.
Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a negative power of the singular perturbation parameter ε [epsilon] instead of an exponential power. It is also proved that the zero level sets of the numerical solutions of the Allen-Cahn equation and the Cahn-Hilliard equation approximate the mean curvature flow and the Hele-Shaw flow respectively. Numerical experiments are carried out to verify the theoretical results and to compare the zero level sets of the numerical solutions and the geometric motions.
Part Two focuses on finite element approximations of stochastic geometric PDEs including the phase field formulation of a stochastic mean curvature flow and the level set formulation of the stochastic mean curvature flow. Both formulations give PDEs with gradient-type multiplicative noises. We establish PDE energy laws and the Hölder [Holder] continuity in time for the exact solutions. Moreover, optimal error estimates are derived, and various numerical experiments are carried out to study the interplay of the geometric evolution and gradient-type noises.
Part Three studies finite element methods for a quasi-static model of poroelasticity, which is a fluid-solid interaction multiphysics system at pore scale. We reformulate the original multiphysics system into a new system which explicitly reveals the diffusion process and has a built-in mechanism to overcome the locking phenomenon . Fully discrete finite element methods are proposed for approximating the new system. We derive a discrete energy law and optimal error estimates for our finite element methods. Numerical experiments are also provided to verify the theoretical results and to confirm that the locking phenomenon has indeed been overcome
Strong convergence rates of an explicit scheme for stochastic Cahn-Hilliard equation with additive noise
In this paper, we propose and analyze an explicit time-stepping scheme for a
spatial discretization of stochastic Cahn-Hilliard equation with additive
noise. The fully discrete approximation combines a spectral Galerkin method in
space with a tamed exponential Euler method in time. In contrast to implicit
schemes in the literature, the explicit scheme here is easily implementable and
produces significant improvement in the computational efficiency. It is shown
that the fully discrete approximation converges strongly to the exact solution,
with strong convergence rates identified. To the best of our knowledge, it is
the first result concerning an explicit scheme for the stochastic Cahn-Hilliard
equation. Numerical experiments are finally performed to confirm the
theoretical results.Comment: 24 pages, 3 figure
Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation
We study a class of fully-discrete schemes for the numerical approximation of
solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and
driven by additive noise. The spatial (resp. temporal) discretization is
performed with a spectral Galerkin method (resp. a tamed exponential Euler
method). We consider two situations: space-time white noise in dimension
and trace-class noise in dimensions . In both situations, we prove
weak error estimates, where the weak order of convergence is twice the strong
order of convergence with respect to the spatial and temporal discretization
parameters. To prove these results, we show appropriate regularity estimates
for solutions of the Kolmogorov equation associated with the stochastic
Cahn--Hilliard equation, which have not been established previously and may be
of interest in other contexts
Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit
Abstract. We consider the stochastic Cahn-Hilliard equation with additive noise term
that scales with the interfacial width parameter ε. We verify strong error
estimates for a gradient flow structure-inheriting time-implicit discretization, where ε
only enters polynomially; the proof is based on higher-moment estimates for iterates, and
a (discrete) spectral estimate for its deterministic counterpart. For Îł sufficiently large,
convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka
problem in the sharp-interface limit ε → 0 is shown. These convergence results are partly
generalized to a fully discrete finite element based discretization.
We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution
in the sharp-interface limit. For this purpose we compare the simulations with those from a
fully discrete finite element numerical scheme for the (stochastic) Mullins-Sekerka problem.
The computational results indicate that the limit for γ ≥ 1 is the deterministic problem,
and for Îł = 0 we obtain agreement with a (new) stochastic version of the Mullins-Sekerka
problem
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