Virtual Element Method For Solving Elliptical Interface Problems Based On Adaptive Interface Fitting Mesh

The interface problem is an important mathematical model problem in science and engineering computing.Such problems often contain complex interfaces and the physical quantity with discontinuities across the interface.This brings great difficulties to accurately represent the interface and find the physical quantity near the interface.The virtual element method is a generalization of the traditional finite element on polygonal and polyhedral meshes.Compared with the traditional finite element method,it has better mesh adaptability and numerical stability,and can greatly reduce the requirements on the mesh quality.In this paper,we designed an adaptive interface fitting mesh generation algorithm based on the quadtree back-ground mesh.This algorithm can quickly and adaptively generate a better quality interface based on the interface curvature.But we don't need to know the true cur-vature,and we used a curvature estimation algorithm to mark mesh cells with too large curvature.Near the interface,we introduced the polygon meshes instead of common triangle meshes to avoid the poor quality problem.Based on the adaptive fitting polygonal mesh,we used the virtual element method to solve the elliptic in-terface problem.Finally,we implemented the method based on the Fealpy package.and the numerical experiments are made to show that the method is effective.