A hybrid FETI-DP method for non-smooth random partial differential equations
- Eigel, Martin
- Gruhlke, Robert
2010 Mathematics Subject Classification
- 35R60 47B80 60H35 65C20 65N12 65N22 65N55 65J10
- Elliptic pde, partial differential equations with random coefficients, domain decomposition, FETI, non-smooth, uncertainty quantification, stochastic finite element method
A domain decomposition approach exploiting the localization of random parameters in high-dimensional random PDEs is presented. For high efficiency, surrogate models in multi-element representations are computed locally when possible. This makes use of a stochastic Galerkin FETI-DP formulation of the underlying problem with localized representations of involved input random fields. The local parameter space associated to a subdomain is explored by a subdivision into regions where the parametric surrogate accuracy can be trusted and where instead Monte Carlo sampling has to be employed. A heuristic adaptive algorithm carries out a problem-dependent hp refinement in a stochastic multi-element sense, enlarging the trusted surrogate region in local parametric space as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration in the involved surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on sub-domains, e.g. in a multi-physics setting, or when the Karhunen-Loeve expansion of a random field can be localized. The efficiency of this hybrid technique is demonstrated with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and non-trusted sampling regions.