Bulk-surface virtual element method for systems of coupled bulk-surface PDEs in two-space dimensions
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In this manuscript we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of coupled systems of elliptic and parabolic bulk-surface partial differential equations (BSPDEs) in two dimensions. To the best of the authors' knowledge, the proposed method is the first application of the virtual element method [Beirao da Veiga et al., M3AS (2013)] to BSPDEs. The BSVEM is based on the discretisation of the bulk domain into polygonal elements with arbitrarily many edges, rather than just triangles. The polygonal approximation of the bulk induces a piecewise linear approximation of the one-dimensional surface (a curve). The bulk-surface finite element method on triangular meshes [Elliott, Ranner, IMAJNA (2013)] is a special case of the proposed method. The present work contains several contributions. First, we show that the proposed method has optimal 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 1/4 is required by the combined effect of surface curvature and polygonal elements. Our analysis shows, as a by-product, that first-degree virtual elements for bulk-only PDEs retain optimal convergence in the presence of surface curvature. In carrying out the analysis we provide novel theoretical tools for the analysis of curved boundaries and non-constant boundary conditions, such as the Sobolev extension and a special inverse trace operator. We show that general polygons can be exploited to reduce the computational complexity of the matrix assembly. Moreover, we present an optimised matrix implementation that can also be exploited in the pre-existing special case of bulk-surface finite elements on triangular meshes [Elliott, Ranner, IMAJNA (2013)]. Numerical examples illustrate our findings and experimentally show the optimal convergence rates in space and time.
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