New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method (Communications in Computational Physics, 2023)
New superconvergent structures are proposed and analyzed for the finitevolume element (FVE) method over tensorial meshes in general dimension $d$ (for $d≥2$);we call these orthogonal superconvergent structures. In this framework, one has the freedom to choose the superconvergent points of tensorial $k$-order FVE schemes (for $k≥3$).This flexibility contrasts with the superconvergent points (such as Gauss points andLobatto points) for current FE schemes and FVE schemes, which are fixed. The orthogonality condition and the modified M-decomposition (MMD) technique that aredeveloped over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of theFVE schemes. Numerical experiments are provided to validate our theoretical results.
Keywords: Superconvergence; finite volume; orthogonality condition; tensorial mesh; rectangular mesh.