ERROR ESTIMATION AND ADAPTIVITY FOR STOCHASTIC COLLOCATION FINITE ELEMENTS PART II: MULTILEVEL APPROXIMATION

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 & 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 & Xu, SIAM J. Sci. Comp., 44 (2022), A3393–A3412). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, are discussed herein. The results demonstrate that the optimal convergence rates can be achieved, but only when solving specific types of problems. The codes used to generate the numerical results are available on GitHub.