A Two‐Level Galerkin Reduced Order Model for Nonlinear Problems

We propose, analyze, and investigate numerically a novel two‐level Galerkin reduced order model (2L‐ROM) for the efficient and accurate numerical simulation of nonlinear problems. In the first step, a relatively low‐dimensional nonlinear system is solved. In the second step, the given nonlinear equations are linearized around the solution found in the first step, and the linearized problem is solved. Although, the new 2L‐ROM can be used for general nonlinear equations, in our theoretical investigation, we consider the steady Navier–Stokes equations. We prove that, for appropriate parameters, the 2L‐ROM error bound is of the same order as the error bound for the standard 1L‐ROM. Furthermore, we develop 2L‐ROM parameter scalings that balance the 2L‐ROM's numerical accuracy and computational efficiency. In our numerical investigation, we consider the Burgers equation, which is a simplified computational setting that is often used in the ROM assessment. Our numerical investigation shows that the new 2L‐ROM significantly decreases (by a factor of 2 and even close to 3) the 1L‐ROM computational cost, without compromising its numerical accuracy. Furthermore, we show that the 2L‐ROM parameters that yield the best numerical results are similar to those obtained from the theoretical scalings.