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Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. (English) Zbl 1068.65122

The piecewise linear conforming finite element solution of the Poisson equation approximates on uniform meshes the interpolant to a higher-order then the solution itself. This type of superclose property is studied for the canonical interpolant defined by the nodal functionals of several nonconforming finite elements of the lowest order. Explicit examples are presented, which show that some nonconforming finite elements do not admit the superclose property.
Two nonconforming finite elements satisfying the superclose property are investigated in detail. Applying the postprocessing technique, one can also state a superconvergence property for the discretization error of the post-processed discrete solution to the solution itself. It is also shown that an extrapolation technique leads to an additional improvement of the accuracy of the finite element solution.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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