Relationship between quality of basis for surface approximation and the effect of applying preconditioning strategies to their resulting linear systems

In this paper, we establish a relationship between preconditioning strategies for solving the linear systems behind a fitting surface problem using the Powell–Sabin finite element, and choosing appropriate basis of the spline vector space to which the fitting surface belongs. We study the problem of determining whether good (or effective) preconditioners lead to good basis and vice versa. A preconditioner is considered to be good if it either reduces the condition number of the preconditioned matrix or clusters its eigenvalues, and a basis is considered to be good if it has local support and constitutes a partition of unity. We present some illustrative numerical results which indicate that the basis associated to well-known good preconditioners do not have in general the expected good properties. Similarly, the preconditioners obtained from well-known good basis do not exhibit the expected good numerical properties. Nevertheless, taking advantage of the established relationship, we develop some adapted good preconditioning strategies which can be associated to good basis, and we also develop some adapted and not necessarily good basis that produce effective preconditioners.