A fast interface solver for the biharmonic Dirichlet problem on polygonal domains
- Khoromskij, Boris N.
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 31A30 35J40 65N12 65N30
- Biharmonic equation, boundary integral operators, fast elliptic solvers, interface operators, matrix compression, preconditioning
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 Poincaré-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 Poincaré-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.
- Num. Math. 78 (1998), No.4, pp. 577-596.