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Analysis of the heterogeneous multiscale method for parabolic homogenization problems. (English) Zbl 1129.65067
The authors discuss the heterogeneous multiscale method (HMM) for various parabolic initial-boundary problems with a small parameter expressing the multiscale nature of the problems. Applications within the paper’s scope are found in models of flow in porous media and mechanical properties of composite materials. The resulting problems may be either linear or nonlinear. The HMMs are general methods for designing sublinear algorithms by exploiting the scale separation of the problems. The macroscopic scheme is chosen to be a piecewise linear finite element method, and time integration is performed by the backward Euler scheme. It is shown that the HMM is stable whenever the macroscopic solver is, and the overall error between the HMM solution and the homogenized solution is controlled by the accuracy of the macroscopic solver and the consistency error e(HMM), which arises from the estimate of the macroscopic data from the microscopic model. The error in a weighted space-time Sobolev norm is shown to be linear in the temporal step size and quadratic in the macroscopic spatial mesh width, plus a contribution from e(HMM). The latter satisfies different estimates depending on the problem setting.

MSC:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35K05Heat equation
35K55Nonlinear parabolic equations
35B27Homogenization; equations in media with periodic structure (PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
76M10Finite element methods (fluid mechanics)
74S05Finite element methods in solid mechanics
74E30Composite and mixture properties
74S05Finite element methods in solid mechanics
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