A Cascadic Multigrid Method for Semilinear Elliptic Equations (Journal of Computational Mathematics, 2018)
This paper introduce a cascadic multigrid method for solving semilinear elliptic equationsbased on a multilevel correction method. Instead of the common costly way of directlysolving semilinear elliptic equation on a very fine space, the new method contains somesmoothing steps on a series of multilevel finite element spaces and some solving steps tosemilinear elliptic equations on a very coarse space. To prove the efficiency of the newmethod, we derive two results, one of the optimal convergence rate by choosing the appropriatesequence of finite element spaces and the number of smoothing steps, and the otherof the optimal computational work by applying the parallel computing technique. Moreover,the requirement of bounded second order derivatives of nonlinear term in the existingmultigrid methods is reduced to a bounded first order derivative in the new method. Somenumerical experiments are presented to validate our theoretical analysis.
Keywords: Semilinear elliptic equation; Parallel computing; Cascadic multigrid; Multilevel correction; Finite element method.