A Weakly Over-penalized Symmetric Interior Penalty Method for the Biharmonic Problem

We study a weakly over-penalized symmetric interior penalty method for the biharmonic problem that is intrinsically parallel. Both a priori error analysis and a posteriori error analysis are carried out. The performance of the method is illustrated by numerical experiments. 1. Introduction. Recently, it was noted in [9] that the Poisson problem can be solved by a weakly over-penalized symmetric interior penalty (WOPSIP) method [10, 12, 25] with high intrinsic parallelism. The WOPSIP method satisfies the same error estimates as the standard P 1 finite element method and also the same condition number estimates after precondition-ing. Furthermore, there exist two orderings (edge-wise and element-wise) of the degrees of freedom (dofs) so that the stiffness matrix for the WOPSIP method is the sum of two matrices , each of which is block diagonal with respect to one of these two orderings. In fact, the matrix representing the piecewise Dirichlet form has 3 × 3 diagonal blocks with respect to the element-wise ordering of the dofs, while the matrix representing the jumps across edges has 1 × 1 or 2 × 2 diagonal blocks in the edge-wise ordering. The simple preconditioner is also block diagonal with 1 × 1 or 2 × 2 blocks in the edge-wise ordering of the dofs. These properties of the WOPSIP method make it an attractive candidate for iterative solvers for the Poisson problem. In this paper, we extend the WOPSIP approach to fourth order problems and develop a method that also has high intrinsic parallelism. For simplicity, we consider the biharmonic problem on a bounded polygonal domain Ω ⊂ R 2. Let f ∈ L 2 (Ω). A weak form of the biharmonic problem is to find u ∈ H