Validated Continuation for Equilibria of PDEs

One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone.