Abstract
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.
Highlights
This paper is devoted to the analysis and validation of an adaptive heterogeneous multiscale finite element method (HMM) for solving nonlinear monotone elliptic problems with fast oscillations, i.e. we are looking for the solution u of the following type of equations:−∇ · A (x, ∇u ) = f in Ω, (1) u = 0 on ∂Ω.Here, A (x, ·) is a monotone function, whereas A (·, ξ) contains a fast, microscopic behaviour
In this work we introduce and analyse a new adaptive PetrovGalerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations
We state an a-posteriori error estimate for the heterogeneous multiscale finite element method introduced in Definition 2.4, where our focus is on the L2-error between HMM approximation uH,h and the δ- 0-averaged limit solution uc
Summary
This paper is devoted to the analysis and validation of an adaptive heterogeneous multiscale finite element method (HMM) for solving nonlinear monotone elliptic problems with fast oscillations, i.e. we are looking for the solution u of the following type of equations:. We do give the first a posteriori error estimate for nonlinear monotone problems, we derive this estimate in a very general homogenization setting that does not assume periodicity or stochastic ergodicity To our knowledge this is the first such result for numerical multiscale methods in general, without explicit knowledge of the homogenized operator. Outline: In Section 2 we introduce the heterogeneous multiscale finite element method for monotone operators This method can be used to efficiently determine the homogenized solution of a nonlinear elliptic multiscale problem with -periodic coefficients (see [25] for comparison). To formulate the numerical multiscale method, we introduce the following notation
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