New smoother to enhance multigrid-based methods for Bratu problem

In this paper, we present more investigations of the numerical solution of the 2D Bratu equation to obtain the second solution on the upper branch by Multigrid. Classical smoothers such as Gauss-Seidel and weighted Jacobi have proven ineffective for obtaining the second solution due to the loss of diagonal dominance and the presence of indefinite Jacobian system at some parameter values. In this paper, we modify the Multigrid algorithms by adding and combining some Krylov methods as smoothers to enhance the multigrid efficiency. Though the idea is not new but we could get new enhanced results compared to that presented by Hackbusch [W. Hackbusch, Comparison of different multi-grid variants for nonlinear equations, ZAMM Z. Angew. Math. Mech. 72 (1992) 148-151] and Washio and Oosterlee [T. Washio, C.W. Oosterlee, Krylov subspace acceleration for nonlinear multigrid schemes, Electron. Trans. Numer. Anal. 6 (1997) 271-290].