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On the convergence rate of the cell discretization algorithm for solving elliptic problems. (English) Zbl 0851.65076

The linear selfadjoint multidimensional elliptic problem \[ \sum_{i, j} D_i(a_{ij}(x) D_j u)- a_0(x) u= f \] on a bounded domain \(\Omega\) with nonhomogeneous Dirichlet boundary conditions is approximated by the cell-discretization algorithm, using a partition of \(\Omega\) into Lipschitz cells with piecewise smooth boundaries. The approximate solutions, constructed by the moment collocation, need not belong globally to the \(H^1\)-space, being only weakly continuous across the cell interfaces.
Error estimates are obtained for polynomial bases. Such polynomial implementation is shown to be a nonconforming version of the \(h\)-\(p\) finite element method. The presented experimental examples provide discontinuous approximations that have errors similar to the finite element results.

MSC:

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