Symmetric Homotopy Method for Discretized Elliptic Equations with Cubic and Quintic Nonlinearities

Abstract

Symmetry is analyzed in the solution set of the polynomial system resulted from the eigenfunction expansion discretization of semilinear elliptic equation with polynomial nonlinearity. Such symmetry is inherited from the symmetry of the continuous problem and is rooted in the dihedral symmetry $$D_4$$D4 of the domain. Homotopies preserving such symmetry are designed to efficiently compute all solutions of the polynomial systems obtained from the discretizations for problems with cubic and quintic nonlinearities, respectively. The key points in homotopy construction are the special properties of the polynomial systems arising respectively from the discretizations of $$-\Delta u=u^3$$-Δu=u3 and $$-\Delta u=u^5$$-Δu=u5 in certain eigensubspaces. Such resulting polynomial systems are taken as start systems in the homotopies. Since only representative solution paths need to be followed, a lot of computational cost can be saved. Numerical results are presented to illustrate the efficiency.