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Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle

Spectral approximations on the triangle by orthogonal polynomials are studied in this paper. Optimal error estimates in weighted semi-norms for both the L2 — and H 1 0 -orthogonal polynomial projections are established by using the generalized Koornwinder polynomials and the properties of the Sturm-Liouville operator on the triangle. These results are then applied to derive error estimates for the spectral-Galerkin method for second- and fourth-order equations on the triangle. The generalized Koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral-element approximations in triangular domains.