Mario Ohlberger

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The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (PDEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study(More)
We give a mathematically rigorous definition of a grid for algorithms solving partial differential equations. Unlike previous approaches (Benger 2005, PhD thesis; Berti 2000, PhD thesis), our grids have a hierarchical structure. This makes them suitable for geometric multigrid algorithms and hierarchical local grid refinement. The description is also(More)
One of the most important tools for accelerating comprehensive computations in multi dimensions (in particular in 3-D) is the local adaption of the grid. In order to minimize the computing time the local grid size should be chosen such that the error ||u − uh|| between the exact solution u and the numerical solution uh is less than a given tolerance and(More)
In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations(More)
Modern simulation scenarios require real-time or many-query responses from a simulation model. This is the driving force for increased efforts in Model Order Reduction (MOR) for high dimensional dynamical systems or partial differential equations (PDEs). This demand for fast simulation models is even more critical for parametrized problems. Several(More)
We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Frechet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a(More)
This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation ct+∇·(uf(c))−∇·(D∇c)+λc = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L-norm, independent of the diffusion parameter D. The resulting a(More)