Inf-sup condition for spherical polynomials and radial basis functions on spheres

Abstract

Abstract. Interpolation by radial basis functions and interpolation by polynomials are both popular methods for function reconstruction from discrete data given on spheres. Recently, there has been an increasing interest in employing these function families together in hybrid schemes for scattered data modeling and the solution of partial differential equations on spheres. For the theoretical analysis of numerical methods for the associated discretized systems, a so-called inf-sup condition is crucial. In this paper, we derive such an inf-sup condition, and show that the constant in the inf-sup condition is independent of the polynomial degree and of the chosen point set, provided the mesh norm of the point set is sufficiently small. We then use the inf-sup condition to derive a new error analysis for the hybrid interpolation scheme of Sloan and Sommariva.

Natural Preconditioning and Iterative Methods for Saddle Point Systems

“…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].…”

“…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.…”

Stability and preconditioning for a hybrid approximation on the sphere

“…Typically, uniqueness of the solution and optimal error estimates for saddlepoint problems follow from so-called inf-sup conditions together with appropriate coercivity of the primal operator. For us an essential tool will be the following infsup theorem proved in [13]. In this theorem h X , for a given point set X = X N ⊂ S d , is the mesh norm, defined by…”

“…In this manuscript we concentrate on the stability of the saddle-point formulation of this hybrid scheme, and devise and validate a rapid preconditioned iterative solution method for the solution of the equations from the approximation. We make use of the Brezzi stability and convergence theorem well known in the context of mixed finite elements, along with the new inf-sup condition of [13] to establish convergence of the approximation scheme; and then use the inf-sup condition to obtain an optimal preconditioner.…”

“…In Section 2 we formulate the hybrid approximation scheme and establish notation. Then in Section 3 we describe the inf-sup condition of [13], and use the Brezzi theorem to establish stability and convergence of the scheme. In Section 4 we turn to preconditioning, establishing there the main spectral equivalence result.…”