Multigrid Bifurcation Iteration

We analyze a multigrid technique for the numerical computation of bifurcating solutions of large, sparse systems of nonlinear parameter-dependent equations which arise from discretizations of nonlinear elliptic eigenvalue problems by finite differences or finite elements. The algorithm consists of a nested iteration procedure which employs a multigrid method for singular problems. The underlying iteration scheme related to the Lyapunov–Schmidt method has been widely used for proving the existence of bifurcating branches and also sometimes for numerical calculations. Our multi-grid technique is shown to be a very efficient and theoretically sound tool for computing bifurcating branches of solutions of nonlinear elliptic eigenvalue problems on arbitrarily bounded domains in $\mathbb{R}^2 $. Several numerical examples confirm this.