  We propose a new model reduction framework for problems that exhibit
transport phenomena. As in the moving finite element method (MFEM), our method
employs time-dependent transformation operators and, especially, generalizes
MFEM to arbitrary basis functions. The new framework is suitable to obtain a
low-dimensional approximation with small errors even in situations where
classical model order reduction techniques require much higher dimensions for a
similar approximation quality. Analogously to the MFEM framework, the reduced
model is designed to minimize the residual, which is also the basis for an
a-posteriori error bound. Moreover, since the dependence of the transformation
operators on the reduced state is nonlinear, the resulting reduced order model
is obtained by projecting the original evolution equation onto a nonlinear
manifold. Furthermore, for a special case, we show a connection between our
approach and the method of freezing, which is also known as symmetry reduction.
Besides the construction of the reduced order model, we also analyze the
problem of finding optimal basis functions based on given data of the full
order solution. Especially, we show that the corresponding minimization problem
has a solution and reduces to the proper orthogonal decomposition of
transformed data in a special case. Finally, we demonstrate the effectiveness
of our method with several analytical and numerical examples.
