A COMPACT DIFFERENCE SCHEME FOR THE BIHARMONIC EQUATION IN PLANAR IRREGULAR DOMAINS M. BEN-ARTZI * , I. CHOREV, J.-P. CROISILLE * , AND D. FISHELOV * Abstract. We present a ﬁnite diﬀerence scheme, applicable to general irreg- ular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate Δ 2 Φ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The ﬁnite diﬀerence scheme is Δ 2 Q Φ (0, 0), where Q Φ is the interpolation polynomial. The interpolation polynomial is not uniquely deter- mined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [7] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples. 1. Introduction In this paper we consider a compact ﬁnite diﬀerence scheme for the Dirichlet problem for the biharmonic equation. Φ(x, y)= g 1 (x, y) Φ n (x, y)= g 2 (x, y)  (x, y) ∈ ∂ Ω, (1) Δ 2 Φ(x, y)= f (x, y), (x, y) in Ω. Basically we propose a generalization to an irregular Ω of the well-known nine- point Stephenson scheme [31]. Such a scheme serves as the main building block in the approximation of the two dimensional Navier-Stokes system in the pure streamfunction formulation [5, 6]. Due to the signiﬁcance of the biharmonic operator a large number of methods for discretizing (1) have been proposed. It seems that the majority of these methods are related to the ﬁnite elements methodology (see for example [3, 10, 14, 15, 29] and references therein). However, we concentrate here on ﬁnite diﬀerence methods. In this category it seems that most of the works are limited to the case where Ω is a rectangular domain. In this case our scheme is actually equivalent to [7], where a fast direct solver is proposed. 1.1. Background. Let us review brieﬂy some of the works which are closer in spirit to the ﬁnite diﬀerence approach. In order to avoid the need to deal with a fourth order diﬀerential operator it has Date : June 17, 2009. 2000 Mathematics Subject Classiﬁcation. Primary: 65N06, Secondary: 41A05, 65D05,35A40. Key words and phrases. Biharmonic Problem, Irregular Domain, High Accuracy, Compact Approximations, Finite Diﬀerences. * Partially supported by a French-Israeli scientiﬁc cooperation grant 3-1355. 1