Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

“…This disagrees with the result of the present paper that no correction is needed to prove the same convergence order as on uniform meshes, i.e., supraconvergence takes place. Our kind of analysis works fine in the case of Dirichlet boundary conditions (see the forthcoming paper [6]). We consider here the more complicated boundary conditions of the third kind, which were studied in [3] for s = 2 on nonuniform meshes in rectangular domains.…”

Supraconvergent cell-centered scheme for two dimensional elliptic problems

“…The phenomenon of supraconvergence in more than one space dimension has also been studied in the literature (see e.g. [7,10,11] and [21]). The topic in the context of finite element methods has been treated in the papers [3,4,10,13,16,17,20,22,23,25,33].…”

“…The analysis of the present paper is based on using negative norms. The analysis of supraconvergence with one additional order of convergence in [3] and [7] is more or less explicitly based on the concept of negative norms. In these two papers discrete analogues of the H −1 -norm were considered.…”

“…In these two papers discrete analogues of the H −1 -norm were considered. The concept of negative norms in the analysis of supraconvergence was also used in [4,[7][8][9][10]15] and [17]. The idea in the present paper is to work instead with a discrete version of the H −2 -norm.…”

Discretely Compact Embeddings of Spaces of Cell-Centered Grid Functions

“…In particular, they are important technical tools in order to establish supraconvergence results for schemes on nonuniform meshes (see, e.g., [3][4][5][6]9]). The discrete convergence theory was introduced by Stummel, in [11], and later considered also by Grigorieff and Reinhardt (see, e.g., [7,8,10,12]) among others.…”