Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations

In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266–281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question. 2010 Elsevier Inc. All rights reserved. In [5], Xiu and Shen consider simulations of diffusion problems with uncertainties, which yield the following stochastic diffusion equation: @uðx; y; tÞ @t 1⁄4 r ðjðx; yÞrxuðx; y; tÞÞ þ f ðx; y; tÞ; 8x 2 X; t 2 ð0; T ; ð1Þ uðx; y;0Þ 1⁄4 u0ðx; yÞ; uð ; y; tÞj@X 1⁄4 0; ð2Þ where x = (x1, . . . ,xd) 2 X R, d = 1,2,3 are the spatial coordinates, and y = (y1, . . . ,yN) 2 R, N P 1, is a random vector with independent and identically distributed components. The steady state counterpart of Eqs. (1) and (2) is r ðjðx; yÞrxuðx; y; tÞÞ 1⁄4 f ðx; y; tÞ; uð ; y; tÞj@X 1⁄4 0; 8x 2 X: ð3Þ It is assumed that the random diffusion field takes a simple form jðx; yÞ 1⁄4 j0ðxÞ þ XN