On multi-level bases for elliptic boundary value problems

We study the multi-level method for preconditioning a linear system arising from a Galerkin discretization method of an elliptic boundary value problem of order 2r. The solution is approximated in the spline space S10(Δn) when r = 1 and S3r−1r−1(Δn) or S3r−3r−1(n) when r≥2, where Sdϱ denotes a spline space of smoothness ϱ and degree d, Δn is the nth (either uniform or nonuniform) refinement of a given triangulation Δ0, and n is the triangulation obtained from the nth refinement (either uniform or nonuniform) of a given quadrangulation. We show that we can construct a multi-level basis in these spline spaces which preconditions the linear system so that its condition number is O((n + 1)2).