We study multigrid for solving the stochastic steady-state diffusion problem. We operate under the mild assumption that the diffusion coefficient takes the form of a finite Karhunen-Loéve expansion. The problem is discretized using a finite element methodology using the polynomial chaos method to discretize the stochastic part of the problem. We apply a multigrid algorithm to the stochastic pro...

We consider the numerical solution of time-dependent partial differential equations with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic partial differential equation into a coupled system of deterministic equations by means of a Galerkin projection onto a g...

and this problem can be solved analytically: u (x) = x − sinhx/sinh1. The purpose of starting with this problem is to demonstrate the fundamental concepts and pitfalls in FEM in a situation where we know the correct answer, so that we will know where our approximation is good and where it is poor. In cases of practical interest we will look at ODEs and PDEs that are too complex to be solved ana...

Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function. Although unknowns are still associated to nodes, the function composed by piece-wise polynomials on each element and thus the gradient can be computed element-wise. Finite element spaces can thus be constructed on general triangulations and thi...

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from theTaylor series expansion with unc...

Using the Stochastic Finite Element Method (SFEM) to perform reliability analysis of the free vibration of composite plates with material and fabrication uncertainties has received much attention lately. In this work the stochastic analysis is performed using the First Order Reliability Method (FORM-method 2) and the Second Order Reliability Method (SORM). The basic random variables include lam...

An analysis of developing and fully developed turbulent flow was carried out when different techniques have been used in a zone close to the solid wall. In this paper, wall element technique based on the finite element method (FEM) has been adopted and compared with other techniques for determination of confined turbulent flow with a one equation model used to depict the turbulent viscosity is ...