  In this paper, we develop a nonlinear reduction framework based on our
recently introduced extended group finite element method. By interpolating
nonlinearities onto approximation spaces defined with the help of finite
elements, the extended group finite element formulation achieves a noticeable
reduction in the computational overhead associated with nonlinear finite
element problems. However, the problem's size still leads to long solution
times in most applications. Aiming to make real-time and/or many-query
applications viable, we apply model order reduction and complexity reduction
techniques in order to reduce the problem size and efficiently handle the
reduced nonlinear terms, respectively. For this work, we focus on the proper
orthogonal decomposition and discrete empirical interpolation methods. While
similar approaches based on the group finite element method only focus on
semilinear problems, our proposed framework is also compatible with quasilinear
problems. Compared to existing methods, our reduced models prove to be superior
in many different aspects as demonstrated in three numerical benchmark
problems.
