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Keywords:

  • iterative methods;
  • algebraic multigrid;
  • smoothed aggregation;
  • stochastic PDEs;
  • stochastic Galerkin

SUMMARY

We consider two algebraic multilevel solvers for the solution of discrete problems arising from PDEs with random inputs. Our focus is on problems with large jumps in material coefficients. The model problem considered is that of a diffusion problem with uncertainties in the diffusion coefficients and realization values differing dramatically between different layers in the spatial domain, with the location of the interfaces between the layers assumed to be known. The stochastic discretization is based on the generalized polynomial chaos, and the spatial problem is discretized using conforming finite elements. A multigrid solver based on smoothed aggregation is presented, and numerical experiments are provided demonstrating convergence properties of the multigrid solver. The observed convergence is shown to depend only weakly on the stochastic discretization and, for the discretized stochastic PDE problem, generally mimics that of the corresponding deterministic problem. Copyright © 2014 John Wiley & Sons, Ltd.