Error estimates of fully discrete finite element solutions for the 2D Cahn-Hilliard equation with infinite time horizon

2016 ◽  
Vol 33 (3) ◽  
pp. 742-762 ◽  
Author(s):  
Ruijian He ◽  
Zhangxin Chen ◽  
Xinlong Feng
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Time Horizon ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Discrete Finite Element

scholarly journals Numerical approximation of the stochastic Cahn–Hilliard equation near the sharp interface limit

2021 ◽  
Author(s):  
Dimitra Antonopoulou ◽  
Ĺubomír Baňas ◽  
Robert Nürnberg ◽  
Andreas Prohl
Keyword(s):  
Finite Element ◽  
Gradient Flow ◽  
Sharp Interface ◽  
Convergence In Probability ◽  
Sharp Interface Limit ◽  
Stochastic Version ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Discrete Finite Element

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

scholarly journals Infinite time horizon optimal current control of a stepper motor exploiting a finite element model

2014 ◽  
Vol 62 (4) ◽  
pp. 835-841 ◽  
Author(s):  
J. Bernat ◽  
S. Stępień ◽  
A. Stranz ◽  
G. Szymański ◽  
J.K. Sykulski
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Time Horizon ◽  
Element Model ◽  
Current Control ◽  
Stepper Motor ◽  
Linear Quadratic ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Optimal Current

Abstract An optimal control theory based method is presented aiming at minimizing the energy delivered from source and the power loss in a stepper motor circuit. A linear quadratic current regulator with an infinite time horizon is employed and its appropriateness for this type of a problem explained. With the purpose of improving the accuracy of the control system, the self and mutual inductances of windings are calculated using a finite element model. The numerically computed results are verified experimentally.

Sign in / Sign up

Export Citation Format

Share Document