  In this paper, we focus on the reduced basis methodology in the context of
non-linear non-affinely parametrized partial differential equations in which
affine decomposition necessary for the reduced basis methodology are not
obtained [4, 3]. To deal with this issue, it is now standard to apply the EIM
methodology [8, 9] before deploying the Reduced Basis (RB) methodology. However
the computational cost is generally huge as it requires many finite element
solves, hence making it inefficient, to build the EIM approximation of the
non-linear terms [9, 1]. We propose a simultaneous EIM Reduced basis algorithm,
named SER, that provides a huge computational gain and requires as little as N
+ 1 finite element solves where N is the dimension of the RB approximation. The
paper is organized as follows: we first review the EIM and RB methodologies
applied to non-linear problems and identify the main issue, then we present SER
and some variants and finally illustrates its performances in a benchmark
proposed in [9].
