An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction–diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a f...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2016-04, Vol.436 (2), p.911-943
Main Authors: März, Thomas, Rockstroh, Parousia, Ruuth, Steven J.
Format: Article
Language:English
Subjects:
Blow-up
Closest Point Method
Implicit surfaces
Laplace–Beltrami operator
Singularity
Surface-intrinsic differential operators
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Summary:In this paper we construct a parametrization-free embedding technique for numerically evolving reaction–diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction–diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.11.064