The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift

Patrick Henning; Mario Ohlberger; Patrick Henning; Mario Ohlberger

doi:10.3934/nhm.2010.5.711

Networks and Heterogeneous Media

2010, Volume 5, Issue 4: 711-744. doi: 10.3934/nhm.2010.5.711

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The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift

Patrick Henning ^{1
,},
Mario Ohlberger ^{1
,}

1.
Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster

Received: 01 March 2010 Revised: 01 May 2010
Primary: 35K15, 35B45, 65N30.

This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.
- Advection-diffusion equation,
- HMM,
- multiscale methods,
- Finite Element scheme,
- error estimate
Citation: Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift[J]. Networks and Heterogeneous Media, 2010, 5(4): 711-744. doi: 10.3934/nhm.2010.5.711

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Abstract

This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.

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