scholarly journals Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth plane domains

1984 ◽  
Vol 29 (4) ◽  
pp. 272-285
Author(s):  
Michal Křížek ◽  
Pekka Neittaanmäki
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Smooth Plane

Stable finite element approximation of a Cahn–Hilliard–Stokes system coupled to an electric field

2016 ◽  
Vol 28 (3) ◽  
pp. 470-498
Author(s):  
ROBERT NÜRNBERG ◽  
EDWARD J. W. TUCKER
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Organic Solar Cells ◽  
Stokes System ◽  
Chemical Potential ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Finite Element Approximation ◽  
Strength Parameter ◽  
Element Approximation

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system: $$\begin{align*} \gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u - \nabla \cdot \left( \nabla w \right) & = 0 \,, \quad w= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u) | \nabla \phi |^2\,, \\ \nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad \begin{cases} -\Delta v + \nabla p = \varsigma w \nabla u, \\ \nabla \cdot v = 0, \end{cases} \end{align*}$$ subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈ [−1, 1], and mixed boundary conditions. Here, γ ∈ $\mathbb{R}_{>0}$ is the interfacial parameter, α ∈ $\mathbb{R}_{\geq0}$ is the field strength parameter, Ψ is the obstacle potential, c(⋅, u) is the diffusion coefficient, and c′(⋅, u) denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential, φ is the electro-static potential, and (v, p) are the velocity and pressure. The system has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field and kinetics.

scholarly journals Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

2011 ◽  
Vol 2011 ◽  
pp. 1-24 ◽  
Author(s):  
Guillaume Jouvet ◽  
Jacques Rappaz
Keyword(s):  
Finite Element ◽  
Mixed Boundary ◽  
Stokes Problem ◽  
A Priori ◽  
Mixed Boundary Conditions ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Numerical Convergence ◽  
Pressure Fields ◽  
Ice Velocity

The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.

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