A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains

We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate Δ²Φ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is Δ²QΦ(0,0), where QΦ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303–333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.