7 Febbraio, 2013 11:00

Reduced basis methods in the context of hierarchical model reduction

Kathrin Smetana, Institute of Computational and Applied Mathematics, University of Munster

Aula Seminari - III piano

Many phenomena in fluid dynamics have dominant spatial directions along which the essential dynamics occur. Nevertheless, the processes in the transverse directions are often too relevant for the whole problem to be neglected. For such situations we present a new problem adapted version of the hierarchical model reduction approach. The hierarchical model reduction approach (see [PerErnVen10] and references therein) uses a truncated tensor product decomposition of the solution and hierarchically reduces the full problem to a small lower dimensional system in the dominant directions, coupled by the transverse dynamics. In previous approaches [PerErnVen10] these transverse dynamics are approximated by a reduction
space constructed from a priori chosen basis functions such as trigonometric or Legendre polynomials. We present the hierarchical model reduction-reduced basis approach [OhlSme10] where the reduction space is constructed a posteriori from solutions (snapshots) of appropriate reduced parametrized problems in the transverse directions. Numerical experiments demonstrate that the hierarchical model reduction-reduced basis approach converges exponentially fast with respect to the model order for problems with smooth solutions but also for some test cases where the source term belongs to $C^{0}( Omega)$ only. Run-time experiments verify a linear scaling of the proposed method in the number of degrees of freedom used for the computations in the dominant direction.

contatto: simona.perotto@polimi.it