A fast finite difference method for biharmonic equations on irregular domains Guo Chen ∗ Zhilin Li † Ping Lin ‡ Abstract Biharmonic equations have many applications, especially in fluid and solid mechanics, but difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical example show the effi- ciency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain. 1 Introduction In this paper, we consider a biharmonic equation defined on an irregular domain Ω Δ 2 u(x, y)= f (x, y), (x, y) ∈ Ω, u(x, y)= g 1 (x, y), (x, y) ∈ ∂ Ω, (1.1) u n (x, y)= g 2 (x, y), (x, y) ∈ ∂ Ω, where Δ 2 ≡∇ 4 = ∂ 4 ∂x 4 +2 ∂ 4 ∂x 2 ∂y 2 + ∂ 4 ∂y 4 , (1.2) Ω is a bounded open set in R 2 with a smooth boundary ∂ Ω, u n = ∂u ∂n is the normal derivative of u on ∂ Ω, and n is the unit normal derivative pointing outward, see Fig. 1 for an illustration. * Department of Mathematics, North Carolina State University, Raleigh, NC 27695, e-mail: gchen@unity.ncsu.edu. † Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695, e-mail: zhilin@math.ncsu.edu. ‡ Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, e-mail: matlinp@nus.edu.sg. 1