DCDS-B
Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise
Georgios T. Kossioris Georgios E. Zouraris
Discrete & Continuous Dynamical Systems - B 2013, 18(7): 1845-1872 doi: 10.3934/dcdsb.2013.18.1845
We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate stochastic parabolic problem discretizing the noise using linear splines. Then we construct fully-discrete approximations to the solution of the approximate problem using, for the discretization in space, a Galerkin finite element method based on $H^2-$piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates: for the error between the solution to the problem and the solution to the approximate problem, and for the numerical approximation error of the solution to the approximate problem.
keywords: space derivative of space-time white noise fully-discrete approximations Backward Euler time-stepping Finite element method a priori error estimates.
PROC
Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method
Christos V. Nikolopoulos Georgios E. Zouraris
Conference Publications 2007, 2007(Special): 768-778 doi: 10.3934/proc.2007.2007.768
We consider the following non-local elliptic boundary value problem:

− $w''(x) = \lambda (f(w(x)))/((\eq^1_(-1) f(w(z)) dz)^2) \all x \in$ (−1, 1),
$w'(1) + \alpha w(1) = 0$, $w'$(−1) − $\alpha w$(−1)$ = $0,

where $\alpha$ and $\lambda$ are positive constants and $f$ is a function satisfying $f(s)$ > 0, $f'(s) < 0, f''(s) > 0$ for $s > 0, \eq^\infty_0 f(s)ds < \infty.$ The solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value $\lambda$* of the parameter $\lambda$, at least two solutions for $\lambda < \lambda$* and has no solution for $\lambda > \lambda$*. We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that $\lambda < \lambda$* and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for $f(s) = e^( - s)$ and $f(s) = (1 + s)^( - 2)$. Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for $\lambda < \lambda$*, a similar comparison of the finite element and the finite volume method is presented.

keywords: finite element method quadratic finite elements. finite volume method Non-local elliptic equation
DCDS-B
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves
Pavlos Xanthopoulos Georgios E. Zouraris
Discrete & Continuous Dynamical Systems - B 2008, 10(1): 239-263 doi: 10.3934/dcdsb.2008.10.239
An initial and Dirichlet boundary value-problem for a Klein– Gordon–Schrödinger-type system of equations is considered, which describes the nonlinear interaction between high frequency electron waves and low frequency ion plasma waves in a homogeneous magnetic field. To approximate the solution to the problem a linearly implicit finite difference method is proposed, the convergence of which is ensured by deriving a second order error estimate in a discrete energy norm that is stronger than the discrete maximum norm. The numerical implementation of the method gives a computational confirmation of its order of convergence and recovers known theoretical results for the behavior of the solution, while revealing additional nonlinear features.
keywords: plasma physics linearly implicit finite difference method a priori error estimates.

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