A COMPACT DIFFERENCE SCHEME FOR THE BIHARMONIC EQUATION IN PLANAR IRREGULAR DOMAINS

摘要

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 Delta(2)Phi at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is Delta(2)Q(Phi)(0, 0), where Q(Phi) 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.