4.5 Article

Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

Journal

CALCOLO
Volume 58, Issue 3, Pages -

Publisher

SPRINGER-VERLAG ITALIA SRL
DOI: 10.1007/s10092-021-00418-5

Keywords

Surface PDEs; Geometrically intrinsic operators; Virtual element method; Polygonal mesh; high-order methods

Funding

  1. ERC Project CHANGE from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme [694515]
  2. project HYDROSEM

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A geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells is developed for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The theoretical properties of the classical VEM are extended to the framework by considering the highly anisotropic character of the final discretization, tested extensively on triangular and polygonal meshes with a manufactured solution to verify scheme limitations.
We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

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