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A sparse finite element method with high accuracy. I. (English) Zbl 0989.65134
The authors develop and analyze a new finite element method for second-order elliptic problems. They use rectangular finite elements with $$Q_p= \text{span}\{x^i y^i: 0\leq i,j\leq p\}$$ bases, two-level grids and global superconvergence theory. Combination of these different approaches leads to much fewer degrees of freedom without loss of accuracy on rectangular domains.
The paper includes all mathematical justifications but, unfortunately, it does not report any numerical experiments. It is worth to mention that the authors plan to extend their studies to more general second-order elliptic equations on more general domains using different kinds of meshes.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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