Superconvergence Analysis of Spectral Volume Methods for Two‐Dimensional Diffusion Equations

ABSTRACT This paper is concerned with the superconvergence analysis of spectral volume (SV) methods for 2D diffusion equations over rectangular meshes. The numerical scheme is constructed by rewriting the diffusion equation into an equivalent first‐order system, and then using the SV method to solve the system. Superconvergence property of two classes of SV schemes based on alternating fluxes are investigated, which are designed by using the Gauss points or Radau points of the underlying meshes to construct control volumes. The th superconvergence rate for the error of numerical fluxes at nodes and for the cell average are obtained when piecewise polynomials of degree are used. Furthermore, interior superconvervgence points for both the function value and derivative value approximations are discovered, which are identified as Gauss points or Radau points. Numerical experiments are presented to validate our theoretical findings.