A Conforming Virtual Element Method Based on Unfitted Meshes for the Elliptic Interface Problem

In this paper, we propose a conforming virtual element method based on an unfitted mesh to solve the elliptic interface problem in two dimensions. The intersecting points of the interface and the edges of triangles are considered as additional nodes of the mesh. Thus each interface triangle is regarded as a polygon with more than three vertices. On each interface polygon, we introduce a virtual element satisfying the interface conditions. On each non-interface triangle, we use the usual linear element. Based on a computable projection-like operator, we introduce our discrete scheme. Both the approximation and consistency errors are analyzed rigorously and all the hidden constants do not depend on how the interface intersects with the meshes. The error between the exact and discrete solution is shown to decrease linear with regard to the mesh size. Some numerical experiments are provided to verify the theoretical results.