Difference potentials method for models with dynamic boundary conditions and bulk-surface problems
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Difference potentials method for models with dynamic boundary conditions and bulk-surface problems Yekaterina Epshteyn1 · Qing Xia1 Received: 17 April 2019 / Accepted: 30 May 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations–based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using difference potentials framework, we develop novel numerical algorithms for the approximation of the problems. The constructed algorithms efficiently and accurately handle the coupling of the models in the bulk and on the surface, approximate 3D irregular geometry in the bulk by the use of only Cartesian meshes, employ fast Poisson solvers, and utilize spectral approximation on the surface. Several numerical tests are given to illustrate the robustness of the developed numerical algorithms. Keywords Dynamic boundary conditions · Bulk-surface models · Difference potentials method · Cartesian grids · Irregular geometry · Finite difference · Spectral approximation · Spherical harmonics Mathematics Subject Classification (2010) 65M06 · 65M12 · 65M70 · 35K10
Communicated by: Gunnar J Martinsson Yekaterina Epshteyn
[email protected] Qing Xia [email protected] 1
Department of Mathematics, University of Utah, 155 S 1400 E Rm. 233, Salt Lake City, UT, 84112, USA
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Adv Comput Math
(2020) 46:67
1 Introduction The parabolic models with dynamic boundary conditions and parabolic bulk-surface models can be found in a variety of applications in fluid dynamics, materials science, and biological applications, see for example, [4, 5, 7, 10–12, 15, 16, 19, 22, 24, 26, 28]. In many of these problems, partial differential equations (PDE)–based models are used to capture dynamic phenomena that occur on the surface of the domain and in the bulk/domain. For instance, cell polarizations can be modeled by the switches of Rho GTPases between the active forms on the membrane (surface) and inactive forms in the cytosol (bulk) [12]. Another example is the modeling of the receptor-ligand dynamics [15], to name a few examples here. In the current literature, there are only few numerical methods developed for such problems, and most of the methods are finite-element based. For instance, a novel finite element scheme is proposed and analyzed for 3D elliptic bulk-surface problems in [14], where polyhedral elements are constructed in the bulk region, and the piecewise polynomial boundary faces serve as the approximation of the surface. The method in [14] employs two finite-element spaces, one in the bulk, and one on the surface. See also the review paper [13] on the finite element methods for PDEs on curved surfaces and the references therein. Also, space and time discretizations of 2D h
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