A FAST INTERFACE SOLVER FOR THE BIHARMONIC DIRICHLET PROBLEM ON POLYGONAL DOMAINS

In this paper we propose and analyze an efficient discretization schem e for the boundary reduction of the biharmonic Dirichlet problem on co nvex polygonal domains. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with a Poincare-Steklov operator acting between subspaces of the trace spaces. We then propose a mixed FE discretization (by lin ear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity log epsilon-O-1(N l og(q) N). Here N is the number of degrees of freedom on the underlying boundary, epsilon > 0 is an error reduction factor, q = 2 or q = 3 fo r rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary redu ctions of the biharmonic Dirichlet problem on convex polygonal domains is derived, A numerical example confirms the theory.