A finite element method for computing the bifurcation function for semilinear elliptic BVPs

The method of alternative problems can be used to show that a semilinear elliptic boundary value problem (Lu + g(x, u) = 0 with gu(x, u) bounded below) is equivalent to a finite-dimensional problem (F(c) = 0 ∈ Rd, c ∈ Rd), in the sense that their solution sets, which are not necessarily singletons, are in a one-to-one correspondence. This correspondence is based on a map σ from low-frequency to high-frequency Fourier components of solutions. A numerical method is presented for approximating σ and hence also solutions of the BVP. The method uses finite element approximations and avoids the use of eigenfunction expansions. Existence, uniqueness, and error estimates for the approximations of σ and solutions u are derived.