  In this work we introduce and analyze a new multiscale method for strongly
nonlinear monotone equations in the spirit of the Localized Orthogonal
Decomposition. A problem-adapted multiscale space is constructed by solving
linear local fine-scale problems which is then used in a generalized finite
element method. The linearity of the fine-scale problems allows their
localization and, moreover, makes the method very efficient to use. The new
method gives optimal a priori error estimates up to linearization errors. The
results neither require structural assumptions on the coefficient such as
periodicity or scale separation nor higher regularity of the solution. The
effect of different linearization strategies is discussed in theory and
practice. Several numerical examples including stationary Richards equation
confirm the theory and underline the applicability of the method.
