Extended Virtual Element Method for the Membrane Problem with Field Singularities and Discontinuities

Among the many stabilized Galerkin finite element formulations proposed in the literature, the recently proposed Virtual Element Method (VEM) [1] stands out for its capability of dealing with very general polygonal or polytopal meshes, in which the basis functions are are not known explicitly within the problem domain (hence, virtual). The bilinear form on each element is decomposed into two parts, by means of suitably defined elliptic projectors: a consistent term, exactly reproducing a the first-order polynomial space, and an additional term ensuring stability. In this contribution, we propose a first-order extended virtual element method (X-VEM) to treat singularities and crack discontinuities that arise in the mambrane problem. The approach herein draws from the development of the extended finite element method for fracture problems [2], in which the discrete space is augmented by means of additional basis functions that capture the main features of the exact solution. A similar approach is pursued in the proposed X-VEM formulation with a few notable extensions. To suitably represent singularities and discontinuities in the discrete space, we extend the standard virtual element space with an additional contribution consisting of the product of virtual nodal basis functions with so-called enrichment functions. For discontinuities, the enrichment function is the generalized Heaviside function across the crack and for singularities it is a weakly singular function that satisfies the Laplace equation. For the membrane problem with a discontinuity, we project the virtual basis functions onto affine polynomials over the two partitions of elements cut by the discontinuity. For the membrane problem with a singularity, we devise an extended projector that maps functions lying in the extended virtual element space onto affine polynomials and the enrichment function. Numerical experiments are performed on quadrilateral and polygonal meshes for the problem of an L-shaped domain with a corner singularity and the problem of a cracked membrane under mode III loading. Obtained results show the accuracy and demonstrate optimal rates of convergence in both L2 norm and energy of the proposed method.