Fast computations with the harmonic Poincaré-Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincar'e-Steklov operators in presence of nested mesh refinement. For both interior and exterior problems the matrix-vector multiplication for the finite element approximations to the Poincar'e-Steklov operators is shown to have a complexity of the order O(Nreflog3N) where Nref is the number of degrees of freedom on the polygonal boundary under consideration and N = 2-p0 · Nref, p0 ≥ 1, is the dimension of a finest quasi-uniform level. The corresponding memory needs are estimated by O(Nreflog2N). The approach is based on the multilevel interface solver (as in the case of quasi-uniform meshes, see [20]) applied to the Schur complement reduction onto the nested refined interface associated with nonmatching decomposition of a polygon by rectangular substructures