Alexei Bespalov and Catherine E. Powell and David Silvester
SIAM Journal on Numerical Analysis. 2012;50(4):2039-2063.
We construct stochastic Galerkin approximations to the solution of a first-order system
of PDEs with random coefficients. Under the standard finite-dimensional noise assumption,
we transform the variational saddle point problem to a parametric deterministic one.
Approximations are constructed by combining mixed finite elements on the computational
domain with $M$-variate tensor product polynomials. We study the inf-sup stability
and well-posedness of the continuous and finite-dimensional problems, the regularity
of solutions with respect to the $M$ parameters describing the random coefficients,
and establish a priori error estimates for stochastic Galerkin finite element approximations.