Many phenomena in nature have dominant spatial directions along which the essential dynamics occur. Nevertheless, the processes in the transverse direction(s) are often too relevant for the whole problem to be neglected. For such situations we present a new hierarchical model reduction approach for elliptic equations, where the full problem is replaced by a one dimensional model in the dominant direction that is enriched by suitable basis functions capturing the dynamics in the transverse direction. These basis functions are solutions of lower dimensional problems deduced from the original one and are constructed by means of the reduced basis technology. We derive an a posteriori error estimate that is used for an efficient construction of these basis functions. Numerical experiments show that only few basis functions on the transverse direction suffice to get a good approximation.

@TechReport{OS10,
  author       = {Ohlberger, M. and Smetana, K.},
  title        = {A new problem adapted hierarchical model reduction technique based on reduced basis methods and dimensional splitting},
  institution  = {FB 10, University Muenster},
  number       = {03/10},
  month        = {december},
  year         = {2010},
  type         = {Preprint},
  keywords     = {model reduction},
  url          = \{/2010/OS10},
}