Numerical homogenization methods for parabolic monotone problems

In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A-stable or explicit stabilized methods). We discuss the construction and the analysis of such methods for a range of problems, from linear parabolic problems to nonlinear monotone parabolic problems in the very general Lp(W1,p) setting. We also show that under appropriate assumptions, a computationally attractive linearized method can be constructed for nonlinear problems.