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Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Runchang Lin;  Zhimin Zhang — 2009

Applications of Mathematics

Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Xu Yin;  Waixiang Cao;  Zhimin Zhang — 2024

Applications of Mathematics

We present a unified approach to studying the superconvergence property of the spectral volume (SV) method for high-order time-dependent partial differential equations using the local discontinuous Galerkin formulation. We choose the diffusion and third-order wave equations as our models to illustrate approach and the main idea. The SV scheme is designed with control volumes constructed using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes...

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