We describe a “discretize-then-relax” strategy to globally minimize integral functionals over functions 𝑢𝑢 in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on 𝑢𝑢 and its derivatives, even if it is nonconvex. The “discretize” step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size ℎℎ of the finite element mesh. The “relax” step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order 𝜔𝜔. We prove that, as 𝜔→∞𝜔→∞ and ℎ→0, solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain 𝐿𝑝𝐿𝑝 norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.