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This paper provides extensions of an element agglomeration AMG method to nonlinear elliptic problems discretized by the finite element method on general unstructured meshes. The method constructs coarse discretization spaces and corresponding coarse nonlinear operators as well as their Jacobians. We introduce both standard (fairly quasi-uniformly coarsened) and non-standard (coarsened away) coarse meshes and respective finite element spaces. We use both kind of spaces in FAS type coarse subspace correction (or Schwarz) algorithms. Their performance is illustrated on a number of model problems. The coarsened away spaces seem to perform better than the standard spaces for problems with nonlinearities in the principal part of the elliptic operator.

In this paper, we present results demonstrating the effectiveness of a sensitivity analysis approach to uncertainty quantification of a variably saturated flow model. The basis for our method is a software system which simultaneously solves for solutions of large-scale nonlinear systems of equations and the sensitivity of the solutions to selected parameters. We present test cases showing the effects on the relative uncertainty of pressure due to heterogeneity in the absolute permeability and to differences in parameterizing the Van Genuchten curve soil parameters, α and n .