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Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

Discrete and Continuous Dynamical Systems - Series S , Volume 8, Number 1, page 119--150 - january 2015
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In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the L2-error between the HMM approximation and the solution of the macroscopic limit equation. The a posteriori error estimate is obtained in a general heterogeneous setting with scale separation without assuming periodicity or stochastic ergodicity. The applicability of the method and the usage of the a posteriori error estimate for adaptive local mesh refinement is demonstrated in numerical experiments. The experimental results underline the applicability of the a posteriori error estimate in non-periodic homogenization settings.

Références BibTex

@Article{HO15,
  author       = {Henning, P. and Ohlberger, M.},
  title        = {Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems},
  journal      = {Discrete and Continuous Dynamical Systems - Series S },
  number       = {1},
  volume       = {8},
  pages        = {119--150},
  month        = {january},
  year         = {2015},
  keywords     = {A posteriori estimate, HMM, monotone operator, multiscale methods. },
  doi          = {10.3934/dcdss.2015.8.119 },
  url          = \{/2015/HO15},
}

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