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., 2013] in the bulk domain to a surface finite element method [Dziuk & Elliott, 2013] on the surface. The proposed method, which we coin the Bulk-Surface Virtual Element Method (BSVEM) includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes [Madzvamuse & Chung, 2016]. The method exhibits second-order convergence in space, provided the exact solution is $H^{2+1/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{1}{4}$ is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an $L^2$-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator [Elliott & Ranner, 2013] 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 & Chung, 2016]. Three numerical examples illustrate our findings.
Virtual element methods for a class of fully nonlinear elliptic PDEs
CEREMADE, CNRS, Université Paris–Dauphine, Université PSL（CEREMADE, 法国国家科学研究中心, 巴黎第九大学, 巴黎文理研究大学）；Mathematics Area, International School for Advanced Studies (SISSA)（数学系, 高等研究院）；Mathematics Institute, University of Warwick（华威大学数学研究所）；Department of Mathematics and Institute for Physical Science and Technology, University of Maryland（马里兰大学数学系与物理科学与技术研究所）
Analysis and virtual element discretisation of a Stokes/Biot--Kirchhoff bulk--surface model
Dipartimento di Matematica e Applicazioni, Università degli studi di Milano Bicocca（米兰比可卡大学数学与应用科学系）；SERENA Project-Team, Centre Inria de Paris（巴黎Inria中心SERENA项目组）；CERMICS, ENPC, Institut Polytechnique de Paris（巴黎理工学院国立桥路学校CERMICS）；School of Mathematics, Monash University（蒙纳士大学数学学院）；Universidad Adventista de Chile（智利基督复临安息日会大学）
Non-degenerate Eulerian finite element method for solving PDEs on surfaces
Institute of Numerical Mathematics, Russian Academy of Sciences（俄罗斯科学院计算数学研究所）；Department of Mathematics, University of Houston（休斯顿大学数学系）；Department of Mechanics and Mathematics, Moscow State University（莫斯科国立大学力学与数学系）
Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems
Department of Mathematics, The Pennsylvania State University（数学系，宾夕法尼亚州立大学）；Department of Mathematics, The Hong Kong University of Science and Technology（数学系，香港科学与技术大学）；Department of Mathematics, University of South Carolina（数学系，南卡罗来纳大学）
