Operator adapted spectral element methods. I: Harmonic and generalized harmonic polynomials. (English) Zbl 0941.65112

The approximation of elliptic differential equations by operator adapted spectral elements is investigated. The adapted systems have superior local approximation properties. The approximation by harmonic polynomials is studied for equations with the Laplace operator. Special attention is paid to the approximation of singular functions that arise in corners. By the theory of Bergman and Vekua the results are extended to general elliptic problems. The partition of unity is used in a numerical example.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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