On fast direct Poisson solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method
This paper dealts with a fast Poisson solver and an iterative Stokes solver whose complexities are respectively \(O(N^ 2\log_ 2 N)\) and \(O(N^{5/2}\log_ 2 N)\) in a two-dimensional rectangular domain (\(N\) is the cutoff number of the polynomial expansion in each direction). Both solvers are based on the Legendre-Galerkin approximation. For a sequence of the discretized Stokes systems the author studies numerically the behaviour of the Babuska-Brezzi inf-sup constant \(\beta_{N, m}\) \((m= N, N- 1, N-2,\dots)\). As is well-known the discretized system admits a unique solution if and only if \(\beta_{N, m}> 0\). The behaviour of these constants are also important because the error estimate is optimal if and only if \(\beta_{N, m}\) are independent of \((N, m)\), and the number of iterations in the conjugate Uzawa method is proportional to \(\beta_{N, m}\). The results presented may serve as a reference for users of spectral methods to choose an appropriate pair \((N, m)\) of discretization spaces in the Stokes problem.
scientific article; zbMATH DE number 740621