  In this paper, we propose an efficient extrapolation cascadic multigrid
(EXCMG) method combined with 25-point difference approximation to solve the
three-dimensional biharmonic equation. First, through applying Richardson
extrapolation and quadratic interpolation on numerical solutions on current and
previous grids, a third-order approximation to the finite difference solution
can be obtained and used as the iterative initial guess on the next finer grid.
Then we adopt the bi-conjugate gradient (Bi-CG) method to solve the large
linear system resulting from the 25-point difference approximation. In
addition, an extrapolation method based on midpoint extrapolation formula is
used to achieve higher-order accuracy on the entire finest grid. Finally, some
numerical experiments are performed to show that the EXCMG method is an
efficient solver for the 3D biharmonic equation.
