A multiscale method for heterogeneous bulk-surface coupling. (English) Zbl 1471.65132
The paper at hand deals with the numerical analysis of of numerical schemes for parabolic problems with dynamic boundary conditions. The authors explain how these models can be understood as limits of bulk-bulk coupling with a thin outer domain. Following [I. Lasiecka, Mathematical control theory of coupled PDEs. Philadelphia, PA: SIAM (2002; Zbl 1032.93002)] the paper considers the problem as a coupled problem where bulk- and surface dynamics are coupled via the boundary. The authors consider the weak formulation and the interpretation of the problem as a partial differential algebraic equation that gives rise to a saddle point structure. The authors design a class of inf-sup stable mixed finite element schemes that allows for independet discretisation in the bulk and on the surface.
This decomposition is particularly beneficial if bulk and surface dynamics have different characteristic lengths scales. This is illustrated by applying multiscale schemes on the surface and standard Lagrangian schemes in the interior. Convergence of a semi-spatially-discrete scheme is proven as well as explicit convergence rates for low-regularity solutions. These rates are independent of the oscillatory behavior of the heterogeneities. Thus, coarse discretization parameters that do not resolve the fine scales can be employed.
This decomposition is particularly beneficial if bulk and surface dynamics have different characteristic lengths scales. This is illustrated by applying multiscale schemes on the surface and standard Lagrangian schemes in the interior. Convergence of a semi-spatially-discrete scheme is proven as well as explicit convergence rates for low-regularity solutions. These rates are independent of the oscillatory behavior of the heterogeneities. Thus, coarse discretization parameters that do not resolve the fine scales can be employed.
Reviewer: Jan Giesselmann (Darmstadt)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L80 | Numerical methods for differential-algebraic equations |
Citations:
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