Virtual element methods for general linear elliptic interface problems on polygonal meshes with small edges. (15th September 2022) - Tushar, Jai; Kumar, Anil; Kumar, Sarvesh
Abstract: In this article, we discuss and analyze a conforming virtual element discretization with boundary stabilization term proposed in Brenner and Sung (2018) [30] (suitable for small edges that appeared in the mesh generation) to approximate general linear elliptic problems with discontinuous diffusivity coefficient across the interface. One of the critical features of the proposed virtual element method is to allow the generation of fitted meshes which are independent of the location of the interface. The fitted polygonal meshes naturally admit elements with small edges, and standard virtual element methods/finite element methods fail to handle meshes with arbitrary small edges. With the help of certain projection operators, nearly optimal error estimates are established under realistic assumptions and low regularity of the solution. Numerical experiments are conducted to show the proposed method's flexibility, robustness, and accuracy and validate the theoretical rate of convergence. Highlights: A conforming Virtual Element Method (VEM) is employed to approximate general linear elliptic problems with discontinuous diffusivity constant across the interface. The significant challenges in solving such a problem include the low global regularity of the solutions and suitable mesh generation near the interface. In the proposed discretization, it is easy to generate background-fitted meshes independent of the location of the interface. In our mesh generation, we allow arbitrary small edges (naturally occurred in fitted mesh) and develop the stable virtual element scheme by using boundary stabilization proposed by S.C. Brenner et al. [Math. Models Methods Appl. Sci. 28 (7) (2018) 1291–1336] for establishing the nearly optimal error estimates. We stress that the standard stabilization may not be suitable when polygonal meshes involve arbitrary small edges. Sophisticated analysis is carried out to show the well-posedness of the discrete problem and prove nearly optimal rate of convergence with minimum regularity on the continuous solution. Several numerical experiments are conducted to compare the classical and proposed boundary stabilization behavior on different interfaces to show the flexibility and robustness of the method. Furthermore, the scheme's performance is illustrated with large magnitudes of discontinuities across the interface, which confirms the accuracy and theoretical rates of convergence.