Abstract
In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis.
Highlights
Let {Γ (t)}t∈[0,T ] be a family of closed hypersurfaces in Rn+1(n = 1, 2) evolving in time
Our numerical approach will be based on an implicit representation of Γ (t), so that we suppose in what follows that there exists a smooth function φ : Ω × [0, T ] → R such that for 0 ≤ t ≤ T
For later use we introduce for t ∈ [0, T ], r > 0 the sets
Summary
Since the mesh for the discretization of (1) is fitted to the hypersurface Γ (t), a coupling to a bulk equation is not straightforward This difficulty is not present in Eulerian type schemes, in which Γ (t) is typically described via a level set function defined in an open neighbourhood of Γ (t). As in some of the methods described above, the surface quantity u is extended to a bulk quantity satisfying a suitable parabolic PDE in a neighbourhood of Γ (t) and the bulk equation is localized to a thin layer of thickness with the help of a phase field function (see [15] for a corresponding convergence analysis). The latter decays at a rate O( p) for some p < 1, while a coupling between and the grid size h is not discussed
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