  A multilevel adaptive refinement strategy for solving linear elliptic partial
differential equations with random data is recalled in this work. The strategy
extends the a posteriori error estimation framework introduced by Guignard and
Nobile in 2018 (SIAM J. Numer. Anal, 56, 3121--3143) to cover problems with a
nonaffine parametric coefficient dependence. A suboptimal, but nonetheless
reliable and convenient implementation of the strategy involves approximation
of the decoupled PDE problems with a common finite element approximation space.
Computational results obtained using such a single-level strategy are presented
in part I of this work (Bespalov, Silvester and Xu, arXiv:2109.07320). Results
obtained using a potentially more efficient multilevel approximation strategy,
where meshes are individually tailored, are discussed herein. The codes used to
generate the numerical results are available online.
