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In this paper we extend the hierarchical model reduction framework based onreduced basis techniques for the application to nonlinear partial differentialequations. The major new ingredient to accomplish this goal is the introductionof the adaptive empirical projection method, which is an adaptive integrationalgorithm based on the (generalized) empirical interpolation method. Differentfrom other partitioning concepts for the empirical interpolation method weperform an adaptive decomposition of the spatial domain. We project both thevariational formulation and the range of the nonlinear operator onto reducedspaces. Those reduced spaces combine the full dimensional (finite element)space in an identified dominant spatial direction and a reduction space orcollateral basis space spanned by modal orthonormal basis functions in thetransverse direction. Both the reduction and the collateral basis space areconstructed in a highly nonlinear fashion by introducing a parametrized problemin the transverse direction and associated parametrized operator evaluations,and by applying reduced basis methods to select the bases from thecorresponding snapshots. Rigorous a priori and a posteriori error estimators,which do not require additional regularity of the nonlinear operator are provenfor the adaptive empirical projection method and then used to derive a rigorousa posteriori error estimator for the resulting hierarchical model reductionapproach. Numerical experiments for an elliptic nonlinear diffusion equationdemonstrate a fast convergence of the proposed dimensionally reducedapproximation to the solution of the full-dimensional problem. Runtimeexperiments verify a close to linear scaling of the reduction method in thenumber of degrees of freedom used for the computations in the dominantdirection.

modélisation mathématique et analyse numérique

Hierarchical model reduction of nonlinear partial differential equations  based on the adaptive empirical projection method and reduced basis  techniques