Operator adapted spectral element methods I: Harmonic and generalized harmonic polynomials

The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. C ompared with standard polynomials, these operator adapted systems have supe rior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rate s of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is paid to the approximation of singular functions that arise typically in corners. These results for ha rmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximatio n results for Laplace's equation hold true verbatim, if harmonic polynomial s are replaced with generalized harmonic polynomials. The Partition of Unit y Method is used in a numerical example to construct an operator adapted sp ectral method for Laplace's equation that is based on approximating with ha rmonic polynomials locally. Mathematics Subject Classification (1991): 65N3 0, 30E10.