Monotone Virtual Element Methods for Convection-Dominated Problems

Edge-averaged virtual element methods for convection-dominated problems

The standard-conforming FEMs may fail to provide accurate numerical solutions because the upper bounds of errors blow up when the diffusion coefficient converges to zero. Indeed, the standard FEMs' solutions contain spurious oscillations that deteriorate the solutions' quality entirely. Therefore, our research focuses on the discrete maximum principle (DMP) because the DMP satisfied in a numerical method guarantees discrete solutions without spurious oscillations. The monotonicity property is a sufficient condition for the DMP, so we were particularly interested in the edge-averaged finite element (EAFE) method [Xu and Zikatanov, 1999], a linear monotone FEM implemented efficiently.

Virtual element methods (VEMs), a generalization of FEMs, are novel numerical methods for PDEs on general polygonal (and polyhedral) meshes. Using general polygonal meshes provides convenience in generating meshes with complex geometries or interface-fitted meshes. Compared to other methods on polygons, the VEMs do not require complicated forms (or equations) of local basis functions inside the polygons. More precisely, virtual element functions are defined by a finite number of their linear functionals (i.e., DoFs), e.g., function values at the vertices of a polygon. Such DoFs and geometric data of polygons compute all necessary quantities in implementation, which provides efficiency in numerical simulation.

Hence, our research specializes in developing monotone or stabilized VEMs for convection-dominated problems. We defined proper flux approximations in the lowest order H(curl)-conforming virtual element space. Using the flux approximations and dual areas for mass lumping, we developed a monotone VEM on a Voronoi mesh whose dual is a Delaunay triangulation consisting of acute triangles. Accordingly, the monotone VEM produces stable numerical solutions to convection-dominated problems in such a Voronoi mesh even though the mesh size is not small (see the following figure for comparison). In addition, the EAVE method has relatively low computational complexity compared to other stabilized VEMs for convection-dominated problems.

• S. Cao, L. Chen, and S. Lee, Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems, submitted.

• S. Lee, Edge-averaged virtual element methods for convection-diffusion problems, PhD thesis, Department of Mathematics, University of California, Irvine, 2021.