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Difference potentials method for models with dynamic boundary conditions and bulk-surface problems

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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.

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Acknowledgments

The authors wish to thank P. Bowman and M. Cuma for assistance in computing facility, the CHPC at the University of Utah for providing computing allocations. The authors are also grateful to the referees for their most valuable suggestions.

Funding

Yekaterina Epshteyn received partial support of Simons Foundation Grant No. 415673 and of NSF Grant No. DMS-1905463.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 S 1400 E Rm. 233, Salt Lake City, UT, 84112, USA

    Yekaterina Epshteyn & Qing Xia

Authors
  1. Yekaterina Epshteyn
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  2. Qing Xia
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Correspondence to Yekaterina Epshteyn.

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Communicated by: Gunnar J Martinsson

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Epshteyn, Y., Xia, Q. Difference potentials method for models with dynamic boundary conditions and bulk-surface problems. Adv Comput Math 46, 67 (2020). https://doi.org/10.1007/s10444-020-09798-8

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  • DOI: https://doi.org/10.1007/s10444-020-09798-8

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