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Bulk-surface virtual element method for systems of PDEs in two-space dimensions

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posted on 2023-06-09, 22:15 authored by Massimo Frittelli, Anotida MadzvamuseAnotida Madzvamuse, Ivonne Sgura
In this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method (Beirão Da Veiga et al. in Math Models Methods Appl Sci 23(01):199–214, 2013. https://doi.org/10.1051/m2an/2013138) in the bulk domain to a surface finite element method (Dziuk and Elliott in Acta Numer 22:289–396, 2013. https://doi.org/10.1017/s0962492913000056) on the surface. The proposed method, which we coin the bulk-surface virtual element method includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes (Madzvamuse and Chung in Finite Elem Anal Des 108:9–21, 2016. https://doi.org/10.1016/j.finel.2015.09.002). The method exhibits second-order convergence in space, provided the exact solution is H2+1/4 in the bulk and H2 on the surface, where the additional 14 is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an L2-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator (Elliott and Ranner in IMA J Num Anal 33(2):377–402, 2013. https://doi.org/10.1093/imanum/drs022) for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes (Madzvamuse and Chung 2016). Three numerical examples illustrate our findings.

Funding

New predictive mathematical and computational models in experimental sciences; G1949; ROYAL SOCIETY; WM160017

Unravelling new mathematics for 3D cell migration; G1438; LEVERHULME TRUST; RPG-2014-149

InCeM: Research Training Network on Integrated Component Cycling in Epithelial Cell Motility; G1546; EUROPEAN UNION

History

Publication status

  • Published

File Version

  • Published version

Journal

Numerische Mathematik

ISSN

0029-599X

Publisher

Springer

Volume

147

Page range

305-348

Pages

39.0

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2020-11-24

First Open Access (FOA) Date

2021-01-28

First Compliant Deposit (FCD) Date

2020-11-23

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