arXiv subjects

Lubomir Banas

Publications and source records attributed to Lubomir Banas.

4 recordsLinked to original sources

arXiv2025

Robust a posteriori error analysis of the stochastic Cahn-Hilliard equation with rough noise

We derive a posteriori error estimate for a fully discrete adaptive finite element approximation of the stochastic Cahn-Hilliard equation with rough noise. The considered model is derived from the stochastic Cahn-Hilliard equation with additive space-time white noise through suitable spatial regularization of the white noise. The a posteriori estimate is robust with respect to the interfacial width parameter as well as the noise regularization parameter. We propose a practical adaptive algorithm for the considered problem and perform numerical simulations to illustrate the theoretical findings.

math.NA↗

arXiv2020

Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit

We consider the stochastic Cahn-Hilliard equation with additive noise term $\varepsilon^γg\, \dot{W}$ ($γ>0$) that scales with the interfacial width parameter $\varepsilon$. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $\varepsilon^{-1}$ only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $γ$ sufficiently large, convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp-interface limit $\varepsilon \rightarrow 0$ 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 $γ\geq 1$ is the deterministic problem, and for $γ=0$ we obtain agreement with a (new) stochastic version of the Mullins-Sekerka problem.

math.NA↗

arXiv2019

Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise

We study the the sharp interface limit of $\varepsilon$-dependent two dimensional stochastic Cahn-Hilliard equation driven by space-time white noise and conservative noise as $\varepsilon\to 0$. In the case when the noise is sufficiently small, by comparing the solutions to equation (1.1) with the approximation solution constructed in [ABC94], we show that the limit of the solutions is also solutions to the deterministic Hele-Shaw problem.

math.PR↗

arXiv2011

Attractivity, invariance and ergodicity for SDEs on Riemannian manifolds

We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further applied to characterize invariant and ergodic measures for various SDEs on manifolds.

math.PR↗