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A semidiscrete Galerkin scheme for a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates. (English) Zbl 1467.65111

The authors consider a coupled system of elliptic-parabolic equations posed on two separated spatial scales which are convex. Such macroscopic-microscopic model problems occur in gas-liquid mixtures and lead to so-called two-pressure evolution systems. After some collections of functional analysis tools the authors prove the existence and uniqueness of a weak solution for their multicsale problem. They use a classical Galerkin approximation ansatz known from the parabolic PDE theory. But due to the complex model this leads to a very comprehensive study. The theory part is closed by a convergence study of a semi-discrete Galerkin approximation with the aid of two Ritz projections (microscopic and macroscopic). Finally, some a priori convergence rates are given if the solutions in spatial direction are smooth enough. In the practical part the authors consider a time-discretized version of their analyzed semi-discretization and discuss the macroscopic and microscopic solution on a certain set of parameter values.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35K58 Semilinear parabolic equations
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