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Some superconvergence estimates for a Galerkin method for elliptic problems. (English) Zbl 0575.65105
We define a Galerkin method for the Dirichlet problem on a two dimensional rectangular domain, using tensor products of continuous piecewise polynomials. It is shown that at specific points in the domain \((=Jacobi\) points and mesh points), the approximate solution has higher order convergence than the global rates of convergence, i.e., superconvergence phenomena are observed. We note that our superconvergence concept here is in the sense of pointwise convergence and not the average convergence which implies discrete \(L^ 2\) norm sense.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations