Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems
A. Abdulle and G. Vilmart
Abstract.
A fully discrete analysis of the finite element
heterogeneous multiscale method for a class of nonlinear
elliptic homogenization problems of nonmonotone type is proposed.
In contrast to previous results obtained for such problems
in dimension $d\leq 2$ for the $H^1$ norm and for a semi-discrete formulation
[W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121--156],
we obtain optimal convergence results for dimension $d\leq 3$
and for a fully discrete method, which takes into account the microscale discretization.
In addition, our results are also valid for quadrilateral finite elements,
optimal a-priori error estimates are obtained for the $H^1$ and $L^2$ norms, improved estimates are obtained
for the resonance error and the Newton method used to compute
the solution is shown to converge.
Numerical experiments confirm the theoretical convergence rates and
illustrate the behavior of the numerical method for various
nonlinear problems.
Key Words. nonmonotone quasilinear elliptic problem, numerical quadrature, finite elements,
multiple scales, micro macro errors, numerical homogenization.