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Abstract
We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.
Original language  English 

Article number  109692 
Journal  Journal of Computational Physics 
Volume  419 
Early online date  1 Jul 2020 
DOIs  
Publication status  Published  15 Oct 2020 
Keywords
 math.NA
 cs.NA
 65C20, 65C30, 65N35, 65M75
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 1 Finished

Multiscale Modelling of Aerospace Composites
Butler, R. & Scheichl, R.
Engineering and Physical Sciences Research Council
6/01/14 → 5/02/18
Project: Research council