Finite element matrix sequences: The case of rectangular domains

In the present paper, we consider a preconditioning strategy for Finite Element (FE) matrix sequences {An(a)}n discretizing the elliptic problem (Formula Presented) with a(x, y) being a uniformly positive function and v denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG) like methods, we define the preconditioning sequence: {Pn(a)}n, Pn(a) := D̃1/2 n(a)An(1)D̃1/2 n(a), where D̃n(a) is the suitable scaled main diagonal of An(a). In fact, under the mild assumption of Lebesgue integrability of a(x), the weak clustering at the unity of the corresponding preconditioned sequence is proved. Moreover, if a(x, y) is regular enough and if a uniform triangulation is considered, then the preconditioned sequence shows a strong clustering at the unity so that the sequence {Pn(a)}n turns out to be a superlinear preconditioning sequence for {An(a)}n. The computational interest is due to the fact that the computation with An(a) is reduced to computations involving diagonals and two-level Toeplitz structures {An(1)}n with banded pattern. Some numerical experimentations confirm the efficiency of the discussed proposal.