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.

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.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

scholarly journals AN IMPROVED FINITE ELEMENT APPROXIMATION AND SUPERCONVERGENCE FOR TEMPERATURE CONTROL PROBLEMS

2013 ◽  
Vol 18 (5) ◽  
pp. 631-640 ◽  
Author(s):  
Yuelong Tang
Keyword(s):  
Finite Element ◽  
Temperature Control ◽  
Piecewise Linear ◽  
A Priori ◽  
Finite Element Approximation ◽  
Linear Functions ◽  
Control Problems ◽  
Element Approximation ◽  
Adjoint State ◽  
Admissible Controls

In this paper, we consider an improved finite element approximation for temperature control problems, where the state and the adjoint state are discretized by piecewise linear functions while the control is not discretized directly. The numerical solution of the control is obtained by a projection of the adjoint state to the set of admissible controls. We derive a priori error estimates and superconvergence of second-order. Moreover, we present some numerical examples to illustrate our theoretical results.

Sign in / Sign up

Export Citation Format

Share Document