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Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz \(hp\)-dGFEM. (English) Zbl 1295.31004

Considering a star shaped domain \(D\subset \mathbb C\), satisfying certain conditions, the authors investigate the best approximation on \(D\) of a function \(f:D \to \mathbb C\) by complex variable polynomials. Exponential convergence in polynomial degree has been obtained provided that \(f\) is holomorphic in an open neighbourhood of \(\overline D\) [J. M. Melenk, Numer. Math. 84, No. 1, 35–69 (1999; Zbl 0941.65112)]. For this study, the authors are motivated by the effort to obtain convergence estimates for the \(hp\)-version of Trefftz-type discontinuous Galerkin finite element methods for second-order scalar elliptic boundary value problems. The authors also observe that the results proved here can be extended to general second-order elliptic equations by means of Vekua theory (see, for example, [A. Moiola et al., Z. Angew. Math. Phys. 62, No. 5, 779–807 (2011; Zbl 1266.35016)]).

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30E10 Approximation in the complex plane
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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