The Reduced Basis Element Method: Application to a Thermal Fin Problem

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The reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization. The basic idea is to decompose the domain of computation into a series of subdomains (the elements) that are similar to a few reference domains. These reference domains are actually "filled" with reduced basis functional spaces that are mapped to each subdomain together with the geometry. The discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces. Finally, a mortar-type method is applied to glue the various local functions. In this paper we focus on the definition of the reference shapes, and together with theoretical and numerical justifications of the method, we provide a posteriori error analysis tools that allow us to certify the computational results.