Multilevel a posteriori error estimator for greedy reduced basis algorithms

Abstract

The goal of reduced basis algorithms is to provide a relatively small set of functions which can serve as a basis for sufficiently accurate and fast numerical solution of a parametrized problem for any choice of parameters. Such methods are often employed in parameter identification problems detecting, for instance, material qualities in diffusion problems, elasticity, or in Maxwell equations, or in time dependent problems where time plays the role of the parameter. We deal with greedy reduced basis algorithms. An important part of these algorithms is to estimate the difference between the exact solution of a discretized problem and its projection onto the space spanned by a reduced basis. We introduce a new kind of the estimate, which is based on a multilevel splitting of a discretized solution space, and we compare it with a standard estimate based on bounds to coercivity and continuity constants. Two sided guaranteed bounds to the error can be obtained for both methods. Numerical complexity as well as memory consumption of both methods are comparable, while the multilevel method provides a more accurate spatial distribution of the error.