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

A fast interface solver for the biharmonic Dirichlet problem on polygonal domains

In this paper we propose and analyze an efficient discretization scheme for the boundary reduction of the biharmonic Dirichlet problem on convex polygonal domains. First we study mapping properties of biharmonic Poincar'e-Steklov operators. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with the restriction of the Poincar'e-Steklov operator. We then propose a mixed FE discretization (by linear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity log ε-1O(N logq N). Here N is the number of degrees of freedom on the underlying boundary, ε > 0 is an error reduction factor, q = 2 or q = 3 for rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary reductions of the biharmonic Dirichlet problem on convex polygonal domains is derived. A numerical example confirms the theory