On a global superconvergence of the gradient of linear triangular elements
Journal of Computational and Applied Mathematics
Superconvergence of the gradient of Galerkin approximations for elliptic problems
Journal of Computational and Applied Mathematics
Finite difference schemes and partial differential equations
Finite difference schemes and partial differential equations
SIAM Journal on Scientific and Statistical Computing
Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems
Journal of Computational and Applied Mathematics
Superconvergence of the Shortley-Weller approximation for Dirichlet problems
Journal of Computational and Applied Mathematics
Effective condition number for finite difference method
Journal of Computational and Applied Mathematics
On solution uniqueness of elliptic boundary value problems
Journal of Computational and Applied Mathematics
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The finite difference method (FDM) using the Shortley-Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When u ∈ C3(S) (i.e., u ∈ C3,0(S)) and f ∈ C2(S), the superconvergence rate O(h2) of solution derivatives in discrete H1 norms by the FDM is derived for rectangular difference grids, where h is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform. Comparisons are made on the analysis by the maximum principle and the FEM analysis, conversions between the FDM and the linear and bilinear FEMs are discussed, and numerical experiments are provided to support superconvergence analysis made.