===affil2:

===firstname:
Ivana
===firstname4:

===firstname3:

===lastname2:

===keyword1:
AMG
===lastname:
Pultarova
===firstname5:

===affil6:

===lastname3:

===email:
ivana@mat.fsv.cvut.cz
===keyword2:
OTHER
===keyword_other2:
Stochastic Galerkin Method
===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===competition:
no
===affil7:

===firstname7:

===postal:
Ivana Pultarova
Thakurova 7
Praha 6
166 29
Czech Republic, Europe
===firstname6:

===ABSTRACT:
Stochastic Galerkin method is a popular tool for solving differential equations with uncertain data. We consider a second order scalar elliptic partial differential equation 
$$-\nabla\left(a(x,y)\nabla u(x,y)\right)=b(x)$$
with Dirichlet boundary conditions. The coefficient $a(x,y)$ is not determined exactly. We use a truncated Karhunen-Loeve expansion of $a(x,y)$ and with either uniformly or normally distributed of random variables $y_i$. Variable $x$ is a spatial variable. Weak formulation and finite element discretization of the space part of the solution and polynomial chaos expansion 
of the stochastic part of the solution lead to a system of linear equations, the dimension of which is huge. 
Then efficient preconditioning techniques are demanded. 
We introduce a hierarchical multilevel preconditioning method obtained from a splitting of approximation spaces $V$ with respect to stochastic variables. Especially, we deal with approximation of the stochastic part of the solution by complete polynomials of order $P$. We consider a splitting of $V$ into the direct sum of the complete orthogonal polynomials of order $P-1$ and of the rest of $V$. As one of the main results, we prove that for the uniform distribution of $y_i$ (and thus for Legendre orthogonal polynomials), the corresponding Cauchy-Buniakowski-Schwarz constant is bounded by $\sqrt{P/(2P+1)}$
and thus the condition number of the two-by-two block preconditioned matrix of the problem is less than $3+2\sqrt{2}$ for any $P>1$.
===affil3:

===keyword_other1:

===lastname5:

===affilother:

===title:
Preconditioning of stochastic Galerkin method
===firstname2: