A Priori Error Analysis of Stochastic Galerkin Mixed Approximations of Elliptic PDEs with Random Data
Alexei Bespalov and Catherine E. Powell and David Silvester
SIAM Journal on Numerical Analysis. 2012;50(4):2039-2063.
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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.