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An adaptive moving mesh method for geometric evolution laws and bulk-surface PDEs

Banff International Research Station for Mathematical Innovation and Discovery

In this talk I will consider the adaptive numerical solution of a geometric evolution law where the normal velocity of a curve in two-dimensions is proportional to its local curvature as well as a general non-geometric driving force. An interface tracking approach is used which requires the generation of a moving mesh. It is well known for this class of problems that moving mesh nodes purely in the normal direction can lead to numerical instabilities and hence some form of tangential mesh movement is necessary. We have developed an adaptive moving mesh approach to distribute the mesh points in the tangential direction using a moving mesh PDE. It will be shown that the resulting meshes evolve smoothly in time and are well adjusted to resolve areas of high curvature. Experiments will be presented to highlight the improvement in accuracy obtained using the new method in comparison with uniform arc-length mesh distributions. We will also discuss the use of the evolving adaptive curve mesh in the adaptive generation of bulk meshes for the solution of bulk-surface PDEs in time-dependent domains. The moving mesh approach will then be applied to a range of problems in computational biology including image segmentation, cell tracking and the modelling of cell migration and chemotaxis. This is joint work with Micheal Nolan, Christopher Rowlatt (Mathematics and Statistics Department, University of Strathclyde) and Robert Insall (Beatson Institute for Cancer Research, Glasgow).

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: University of Strathclyde

2019-03-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69200

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/