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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:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35K55 Nonlinear parabolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74E30 Composite and mixture properties
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