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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.