Polyhedra abstract domain is one of the most expressive and used abstract domains for the static analysis of programs. Together with Kleene algorithm, it computes precise yet costly program invariants. Widening operators speed up this computation and guarantee its termination, but they often induce a loss of precision, especially for numerical programs. In this article, we present a process to accelerate Kleene iteration with a good trade-off between precision and computation time. For that, we use two tools: convex analysis to express the convergence of convex sets using support functions, and numerical analysis to accelerate this convergence applying sequence transformations. We demonstrate the efficiency of our method on benchmarks.