A Cartesian mesh approach to embedded interface problems using the virtual element method

In this paper, we propose an elegant methodology to treat sharp interfaces that are implicitly defined which does not require (a) enrichment functions, (b) additional linear and bilinear terms such as the inter-element penalty terms as in Nitsche's method, or use of multipliers like Lagrange multiplier, in the weak form for enforcing the jump conditions across the interface, and (c) modification to the standard virtual element method solution space. The background mesh consists of structured quadrilateral elements with each element consisting of eight nodes, namely, the four vertices and the mid-points of four edges. A simple and efficient idea to generate an interface-fitted mesh is discussed where the number of nodes remains invariant, esp., for moving boundary problems. A linear virtual element method approximation is assumed on the fitted mesh. The efficiency and accuracy of the presented technique is demonstrated by solving and verifying the rate of convergence in both L2 norm and H1 semi-norm, for the benchmark problems with interfaces of various geometries and moving interfaces.