Stability and preconditioning for a hybrid approximation on the sphere

Abstract

Abstract. This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived inf-sup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable block-diagonal preconditioner. Numerical experiments with a non-uniform distribution of data points support the theoretical conclusions.

Natural Preconditioning and Iterative Methods for Saddle Point Systems

“…where τ > 0 is a specified constant [64], the inf-sup condition (2.13) is satisfied, with inf-sup constant β that is independent of N but which depends on L, and Theorem 2.2 holds [44,64,Theorem 2]. From experiments it appears that when N = 4000 then β is no smaller than 0.8 for m = 0, 1, L = 5, 10, 15, 20, 25.…”

“…The inf-sup condition and its discrete counterpart play an important role in mixed methods for PDE problems, such as those arising in fluid dynamics [27,Chapter 5], [25,Chapter 3] and solid mechanics [11,Chapter VI]. However, researchers have begun to appreciate the importance of inf-sup conditions in other applications, such as when developing hybrid interpolants on the sphere [44,64].…”

“…An effective natural preconditioner (4.1) was devised by Le Gia, Sloan and Wathen [44] which we now describe. Since a is equivalent to the norm on X N , the preconditioner P 1 should be spectrally equivalent matrix to A and Le Gia, Sloan and Wathen use a domain decomposition preconditioner.…”

“…For the hybrid interpolation method, we consider interpolation at a set of points on the sphere X N = {x j } j=1,...,N . Assuming that the N data points are distinct, we can construct a radial basis approximation space X N with associated native space norm (see, for example, Le Gia, Sloan and Wathen [44] for more details) and a spherical polynomial approximation space M L , also equipped with the native space norm, where L ≥ 0 is the maximum total degree of the polynomials.…”

“…Related preconditioners that are based only on a study of the eigenvalues of the preconditioned matrix have also been proposed [40,43,49]. Norm-based preconditioning also arises in particular applications in PDEs, such as in groundwater flow [12,13,21,33,55,69], Stokes flow [18,20,76,59,74], elasticity [2,16,32,42,60], magnetostatics [52,53] and in the hybrid interpolation scheme on the sphere [44]. We note that Arioli and Loghin [1] also use norm equivalence to investigate appropriate stopping criteria for iterative methods applied to mixed finite element problems.…”