8th. World Congress on Computational Mechanics (WCCM8) 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008) June 30 –July 5, 2008 Venice, Italy A finite difference scheme for the biharmonic equation in planar irregular domains Matania Ben-Artzi 1 ,*Ittai Chorev 1 , Jean-Pierre Croisille 2 and Dalia Fishelov 3 1 Institute of Mathematics The Hebrew University, Jerusalem 91904, Israel mbartzi@math.huji.ac.il, http://www.ma.huji.ac.il/ mbartzi ichorev@gmail.com 2 Department of Mathematics University of Metz, Metz 57045, France croisil@poncelet.univ-metz.fr, http://www.math.univ- metz.fr/∼croisil 3 School of Mathematical Sci- ences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel daliaf@post.tau.ac.il, http://www.math.tau.ac.il/∼daliaf Key Words: Biharmonic Problem, Irregular Domain, High Accuracy, Compact Approximations, Finite Differences. ABSTRACT The accurate resolution of the biharmonic equation (1), is of great importance in many fields of applied mathematics. We focus on generalizing a finite difference scheme, namely the nine points Stephenson scheme [6], to an irregular domain Ω. Its main interest, is that it allows to construct the time-dependent solver, which fits closely to the pure streamfunction formulation of the Navier-Stokes equation [2, 3]. Φ(x, y)= g 1 (x, y) Φ n (x, y)= g 2 (x, y)  (x, y) ∈ ∂ Ω, Δ 2 Φ(x, y)= f (x, y), (x, y) in Ω (1) We discretize (1) by using values of Φ and ∇Φ at grid points. This method does not require any modi- fications near the boundary, as the boundary condition also include the values of ∇Φ. Some variations and a multigrid technique were proposed in [1]. So far, all the algorithms based on the nine-point stencil were only applicable to a rectangular domain. Fix h> 0 and take all grid points (ih, jh), i, j ∈ Z, which are interior to Ω. Most of them are ”calculated nodes”. Other nodes, which are close to ∂ Ω only serve for the geometric construction of the scheme i.e. they do not take approximate values or serve as ghost points. Our scheme is a compact scheme; all approximate values of derivatives use values of Φ, Φ x , Φ y at its immediate neighbors. For each calculated node, we construct suitable neighboring points p 1 , ..., p 8 , which are either calculated nodes or boundary points (see Figure 1). In order to approximate Δ 2 Φ at p 0 we interpolate the data on the stencil p 0 , ..., p 8 (see Figure 2) by a polynomial of degree six. The finite difference scheme, for the approximation of Δ 2 Φ at p 0 = (0, 0) is Δ 2 Q Φ (0, 0), where Q Φ is the interpolation polynomial. The actual connection between the values of Φ, Φ x , Φ y at all nodes uses a fourth order hermitian form.