  Nonlinear multi-scale problems are ubiquitous in materials science and
biology. Complicated interactions between nonlinearities and (nonseparable)
multiple scales pose a major challenge for analysis and simulation. In this
paper, we study the numerical homogenization for multi-scale elliptic PDEs with
monotone nonlinearity, in particular the Leray-Lions problem (a prototypical
example is the p-Laplacian equation), where the nonlinearity cannot be
parameterized with low dimensional parameters, and the linearization error is
non-negligible. We develop the iterated numerical homogenization scheme by
combining numerical homogenization methods for linear equations, and the
so-called "quasi-norm" based iterative approach for monotone nonlinear
equation. We propose a residual regularized nonlinear iterative method, and in
addition, develop the sparse updating method for the efficient update of coarse
spaces. A number of numerical results are presented to complement the analysis
and valid the numerical method.
