Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems
Patrick Henning, Mario Ohlberger · Discrete and Continuous Dynamical Systems - S · 2014
In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finiteelement method (HMM) for monotone elliptic operators with rapid oscillations.In a general heterogeneous setting we prove convergence of theHMM approximations to the solution of a macroscopic limit equation.The major new contribution of this work is an a-posteriori error estimatefor the $L^2$-error between the HMM approximation and the solution of themacroscopic limit equation.The a posteriori error estimate is obtained in a general heterogeneous settingwith scale separation without assuming periodicity or stochastic ergodicity.The applicability of the method and the usage of the a posteriori error estimatefor adaptive local mesh refinement is demonstrated in numerical experiments.The experimental results underline the applicability of the a posteriori errorestimate in non-periodic homogenization settings.