Extended Virtual Element Method for the Laplace Problem with Singularities and Discontinuities

The Virtual Element Method (VEM) [1] is a stabilized Galerkin finite element formulation that is capable of dealing with very general polygonal or polytopal meshes, wherein the basis functions are implicit (virtual) — they are not known explicitly within the problem domain. Suitable projection operators are used to decompose the bilinear form on each element into two parts: a consistent term that reproduces the first-order polynomial space and a correction term that ensures 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 Laplace problem. The approach herein draws from the development of the extended finite element method for fracture problems [2], in which the discrete space is extended by means of additional basis functions that capture the salient 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 augment the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis functions with so-called enrichment functions. For discontinuities, the enrichment function is discontinuous (generalized Heaviside function) across the crack and for singularities it is a weakly singular function that satisfies the Laplace equation. For the Laplace problem with a discontinuity, we project the virtual basis functions onto affine polynomials over the two partitions of an element cut by the discontinuity. For the Laplace problem with a singularity, we devise an extended projector that maps functions that lie in the extended virtual element space onto linear polynomials and the enrichment function. The homogeneous numerical integration method is used to accurately compute integrals with integrands that are discontinuous or are weakly singular. 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 prove the accuracy and demonstrate optimal rates of convergence in both L2 norm and energy of the proposed method.