Constructive techniques for approximating collocation linear systems (Q5934409)

A high order discretization by spectral collocation methods of the elliptic problem \[ A_{A,b}u= -\nabla\bigl[ A(x)\nabla u(x)\bigr]+ b(x)u (x) =C(x)\text{ on }\Omega= (-1,1)^2; \quad u|_{\partial \Omega}=0; \] in two variables \(x=(x^{[1]}, x^{[2]})\) is considered. The methods give rise to a sequence of dense linear systems that are optially preconditioned by using the finite difference matrix sequence \(\{A_n\}_n\) over the non-uniform grid sequence defined via the collocation points. A preconditioning strategy for \(\{A_n\}_n\) is proposed based on the ``approximate factorization'' idea. Several numerical experiments confirm the goodness of the discussed proposal.