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Articles written by Zouraris, G.E.

  1. Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations" SIAM J. Numer. Anal.. 42 (2). 2004. pp. 800--825.

    We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

    Keywords: stochastic elliptic equation ; perturbation estimates ; Karhunen--Loeve expansion ; finite elements ; Monte Carlo method ; $k\times h$-version ; $p\times h$-version ; expected value ; error estimates


  2. Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation" Computer Methods in Applied Mechanics and Engineering. vol. 194. 2005. pp. 1251--1294.