Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature
This paper addresses Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadratures \[ \int_{-1}^1 \phi(x) \,dx\approx \phi'(-1) \omega_{-} + \sum_{j=1}^{N-1} \phi(x_j) \omega_J +\phi'(1)\omega_{+}, \] which are exact for all polynomials of degree \(2N-1\) and whose interior nodes are zeros of the quasi-orthogonal polynomials formed by linear combination of Jacobi polynomials. GLLB quadratures are particularly important in spectral approximations of second-order elliptic problems with Neumann boundary conditions, as well as for many other important problems, such as numerical integration and integral equations with the data of derivatives at the endpoints. Its nodes can be a natural choice of the preassigned interpolation points. The authors continue studies of \textit{A.~Ezzirani} and \textit{A.~Guessab} [Math. Comput. 68, No. 225, 217--248 (1999; Zbl 0916.65020)] who proposed a fast algorithm for computing nodes and weights of Gauss-Birkhoff quadreature with mixed boundary conditions. The results are related to studies of asymptotic behaviors of the following nodes and weights: \( x_j \cong \cos{\frac{(2j+1/2)\pi}{2N+1}}; \,\, \omega_j \cong \frac{\pi}{N} \sin \theta_j,\, \theta_j=\arccos x_j,\, 1\leq j \leq N-1.\) It is to be noted that GLLB differentiation matrices are ill-conditioned. In the present paper, the authors construct finite-difference preconditioning for one-dimensional second-order elliptic equations. The user-oriented implementation and an error analysis of the GLLB pseudospectral methods based on variational formulations are also provided.
scientific article; zbMATH DE number 5639037
scientific article