A surface mesh DG-VEM for elliptic membrane shell model

Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct $C^0$-continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree l, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.