hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity

Abstract

For a linear second order elliptic partial differential operator $A: V → V'$, we consider the boundary value problems $Au=f$ with stationary Gaussian random data $f$ over the dual $V'$ of the separable Hilbert space $V$ in which the solution u is sought. The operator $A$ is assumed to be deterministic and bijective. The unique solution $u= A^-$$^1f $ is a Gaussian random field over $V$. It is characterized by its mean field $E_u$ and its covariance $C_u$ ∈ $V$ ⊗ $V$. For a class of piecewise analytic covariance kernels $C_f$ ∈ $V'$ ⊗ $V'$ for Gaussian data $f$, we prove analytic regularity of the covariance $C_u$ of the Gaussian solution $u$ in families of countably normed spaces. To this end, we investigate shift theorems for the (non-hypoelliptic) deterministic tensor PDEs $(A$ ⊗ $A)C_u = C_f$ proposed in (14) for the covariance $C_u$ The non-hypoelliptic nature of $A$ ⊗ $A$ implies that sing supp($C_u$) is in general strictly larger than sing supp($C_f$) Based on our regularity results, we outline an $hp$-Finite Element strategy from (7,8) to approximate $C_u$ stemming from covariances of stationary Gaussian data $f$. In the second part (8) of this work, we prove that this discretization gives exponential rates of convergence of the $FE$ approximations, in terms of the number of degrees of freedom.