Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space–time white noise

2013 ◽  
Vol 67 ◽  
pp. 243-261 ◽  
Author(s):  
Georgios T. Kossioris ◽  
Georgios E. Zouraris
Keyword(s):  
Finite Element ◽  
White Noise ◽  
Fourth Order ◽  
Space Time ◽  
Finite Element Approximations ◽  
Three Space

scholarly journals Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Three Dimensional ◽  
Space Time ◽  
Symmetry Analysis ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

FINITE ELEMENT ERROR ANALYSIS OF SPACE AVERAGED FLOW FIELDS DEFINED BY A DIFFERENTIAL FILTER

2004 ◽  
Vol 14 (04) ◽  
pp. 603-618 ◽  
Author(s):  
ADRIAN DUNCA ◽  
VOLKER JOHN
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Flow Fields ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Finite Element Approximations ◽  
Three Space ◽  
Element Error

This paper analyzes finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier–Stokes equations with a differential filter. It is shown that [Formula: see text], the error of the filtered velocity [Formula: see text] and the filtered finite element approximation of the velocity [Formula: see text], converges under certain conditions of higher order than [Formula: see text], the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the L2-error of finite element approximations of [Formula: see text] and [Formula: see text] is considered. Numerical tests in two and three space dimensions support the analytical results.

A convergent finite element scheme for a fourth-order liquid crystal model

2020 ◽  
Author(s):  
Stefan Metzger
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Fourth Order ◽  
Liquid Crystalline ◽  
Stokes Equations ◽  
Finite Element Scheme ◽  
Director Field ◽  
Element Scheme ◽  
Finite Element Approximations ◽  
Stable Splitting

Abstract In this manuscript we propose and analyse a fully discrete, unconditionally stable finite element scheme for a recently developed director model for liquid crystalline flows (Metzger, S. (2020) On a novel approach for modeling liquid crystalline flows. Commun. Math. Sci., 18, 359–378). The model consists of nonlinear fourth-order partial differential equations describing the evolution of the director field and Navier–Stokes equations governing the velocity field. We employ a stable splitting approach to reduce the computational complexity by decoupling the update of the director field from the update of the velocity field. We also perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of finite element approximations converges towards a weak solution of the original model. Passing to the limit in the nonlinear terms requires us to derive the strong convergence of the gradient of the director field from uniform bounds for its discrete Laplacian. Furthermore, we present simulations underlining the practicability of the proposed scheme, investigate its convergence properties and discuss the differences between the underlying model and already established Ericksen–Leslie-type models.

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