Uniform convergence for a finite-element discretization of a viscous diffusion equation

We study a space semidiscretization of a viscous diffusion equation, obtained as a singular limit of the viscous Cahn–Hilliard equation by letting the interfacial energy tend to 0. The semidiscrete solution is shown to converge uniformly in time and space to the continuous solution, on finite time intervals, as the discretization parameter h tends to 0 (in space dimension one, two and three). We obtain an optimal error bound in space dimension one assuming only a piecewise Lipschitz regularity on the initial value. This approach allows us to obtain some counterexamples concerning lower and upper bounds of solutions to Cahn–Hilliard equations. Numerical simulations confirm the theoretical results.