Superlinear preconditioners for finite differences linear systems

We consider a preconditioning strategy for finite differences (FD) matrix sequences {An(a,Ω)}n discretizing the elliptic problem (Aau ≡ (-)k∇k[a(x)∇ ku(x)] = f(x), x ∈ Ω, (∂s/∂ν s u(x))|∂ω ≡0, s=0,...,k-1, with Ω being a plurirectangle of Rd, with a(x) being a uniformly positive (nonnegative) Riemann integrable function, and v denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG)-like methods, we consider the preconditioning sequence {P n(a,Ω)}n, Pn(a,Ω) := D̃ n 1/2(a,Ω)An(1,Ω)D̃ n 1/2(a,Ω), where D̃n(a,Ω) is the suitable scaled main diagonal of An(a,Ω). Using embedding arguments and projection matrices, under the mild assumptions on a(x), we show the weak clustering at the unity of the corresponding preconditioned sequence. If a(x) is regular enough, then the preconditioned sequence shows a strong clustering at the unity so that the sequence {Pn(a,Ω)} n turns out to be a superlinear preconditioning sequence for {A n(a,Ω)}n. The computational interest is due to the fact that the solution of a linear system with coefficient matrix A n(a, Ω) is reduced to computations involving diagonals and multilevel structures {An(1,Ω)}n with banded pattern. In turn, the matrix An(1,Ω) can be reinterpreted as a projection of a multilevel banded Toeplitz matrix for which we use multigrid strategies. Some numerical experimentations confirm the efficiency of the discussed proposal and its strong superiority with respect to existing techniques in the case of semielliptic problems.