Error estimates for the bifurcation function for semilinear elliptic boundary value problems
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A semilinear elliptic boundary value problem of the form Lu+gx,u,l=0 and a corresponding discrete problem based on the finite element method are considered. The method of alternative problems is used to reduce the boundary value problem to an equivalent finite-dimensional problem Bc,l=0 . The bifurcation function Bc,l is a vector field on Rd for fixed l . The solutions of the reduced problem are in a one-to-one correspondence with the solutions of the boundary value problem. The method of alternative problems is also applied to reduce the discrete problem to an equivalent lower-dimensional problem. An approximate bifurcation function Bhc,l for the lower-dimensional problem is also defined as a vector field on Rd , whose zeros are in a one-to-one correspondence with the solutions of the discrete problem. Estimates of the differences Bc,l-Bh c,l,Bc c,l-B hcc,l , and Ec,l-E hc,l are derived. Here, Ec,l (resp. Ehc,l ) denotes the value of energy functional associated with the boundary value problem (resp. the discrete problem). Morse decompositions are computed for some classical examples, and their bifurcation diagrams are presented. Results from numerical experiments on the orders of convergence for the difference Bc,l-Bh c,l are presented.