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

Abstract

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.Keywords Closest Point Method · implicit surfaces · surface-intrinsic differential operators · Laplace-Beltrami operator · blow-up · singularity · resolution of singularity · singular differential operator 1 Introduction

Extrinsic Meshless Collocation Methods for PDEs on Manifolds

“…A2. Assume that \scrL \scrM in (1.1) is a second-order strongly elliptic operator whose coefficients are in \scrW Using the restriction map R cp in (2.2), the manifold operator \scrL \scrM can be embedded to \Omega \delta by [34] (3.4)…”

“…They may require analytical transformations of differential operators on the surface to standard ones in Cartesian (or extrinsic) coordinates or projections. Methods in the embedding class avoid such transformations by extending surface PDEs into some embedding spaces in \BbbR d and work with d-dimensional computational domains; see [4,8,18,31,32,33,34,44,46]. One potential disadvantage is the additional computational cost due to the extension, particularly for high codimensions.…”

Kernel-based collocation methods for heat transport on evolving surfaces

“…To solve the target PDE (1) on the evolving surface, a surface finite element method (SFEM) were proposed with the use of weak and variational formulas based on triangular meshes intrinsically defined on surfaces [6,9]. Embedding techniques for static problems can also be extended to deal with surface PDEs posed in higher-dimensional [10,11,12,13,14]. Some domaintype time-dependent FEMs have been implemented in narrow bands containing moving surfaces [7,8,15,16].…”