Multigrid Method For Solving Biharmonic Equation

Biharmonic equation is an elastic thin plate theory originated from continuum mechanics in physics.It is very important to accelerate the solution of the biharmonic equation for many practical problems,such as engineering design,fluid mechanics and so on.This thesis mainly discusses the cascadic multigrid method for solving biharmonic problems.First,the biharmonic equation is discretized by thirteen point difference scheme.By special treatment of the boundary conditions of the outer two-layer points,the difference discrete system of the biharmonic equation is obtained.Then,the existence of the solution of the discrete scheme is proved by using the lower bound of matrix eigenvalues.The truncation error and the asymptotic expansion of the error are derived.Multi grid method is a very effective method to solve elliptic boundary value problems.In this thesis,based on the asymptotic expansion of the finite difference solution,a new extrapolation solution of the finite difference solution on a dense grid is derived by using the finite difference solution on a two-layer coarse grid.The new extrapolation solution is used as the initial value of the iteration on the dense grid,and the conjugate gradient method(CG)is used to solve the sparse positive definite linear equations obtained by the thirteen point difference separation.Based on this,a fast algorithm,cascadic multigrid method(CMG)and extrapolated cascadic multigrid method(EXCMG),is designed to solve the discrete system of the thirteen point difference scheme of the biharmonic equation.The convergence of cascadic multigrid algorithm(CMG)and the error estimation of extrapolation initial value of extrapolation cascadic multigrid algorithm(EXCMG)are discussed respectively.Finally,for four different cases,the biharmonic equation is discretized by the thirteen point difference scheme,and then solved by direct CG,CMG and EXCMG respectively.The numerical results show that the thirteen point difference scheme for the biharmonic equation has second order convergence.Moreover,the error between the initial values of CMG and EXCMG and the true solution reaches the second order of fullness.In particular,the extrapolated initial value of EXCMG is closer to the accurate difference solution.The error between them reaches the third order convergence in the sense of maximum modulus,and the number of iterations is only 1 / 10 of the number of CG iterations,which proves the superiority of EXCMG.