Conditional spatial finite-element rate for stochastic Galerkin

Assume the hypotheses of the stochastic Galerkin theorem with $D\subset {\mathbb{R}}^d$, $d\ge 2$, and suppose the exact solution has the exact finite expansion $u(y)={\sum }_{\alpha \in \mathrm{\Lambda }}{u}_{\alpha }{\psi }_{\alpha }(y),u_{\alpha }\in H^r(D)\cap H_0^1(D),$ for integers $k\ge 0$ and $1\le \mathrm{\ell }\le r\le k+1$. Choose a fixed reference finite element whose polynomial space contains ${\mathbb{P}}_k$ and whose unisolvent degrees of freedom are bounded on $W^{\mathrm{\ell },2}$ of the reference cell; let its reference interpolant reproduce ${\mathbb{P}}_k$ and define local interpolants by affine pullback as in Local interpolation estimates on shape-regular affine meshes. Let $\{{\mathcal{T}}_h\}$ be a shape-regular family of matching affine simplicial meshes as in Matching affine simplicial meshes and shape regularity, with maximum cell diameter $h$, and assume for each mesh that a globally conforming spatial interpolant $I_h$ maps these coefficients into its finite-element space, restricts cellwise to those local interpolants, and preserves the zero boundary condition. These are the generic bounded-degree-of-freedom hypotheses of the local estimate. Then for the stochastic Galerkin solution $U_{\mathrm{\Lambda },h}$, $\mathrm{\parallel }u-U_{\mathrm{\Lambda },h}{\mathrm{\parallel }}_{L_{\rho }^2(H_0^1(D))}\le \frac{a^{\ast }}{a_{\ast }}C{h}^{r-1}{\left({\sum }_{\alpha \in \mathrm{\Lambda }}\mathrm{\mid }{u}_{\alpha }{\mathrm{\mid }}_{H^r(D)}^2\right)}^{1\mathrm{/}2}\mathrm{.}$ The regularity and finite-expansion assumptions are explicit; this is not a general spatial rate for arbitrary random solutions.