Large time step maximum norm regularity of L-stable difference methods for parabolic equations

Abstract

We establish almost sharp maximum norm regularity properties with large time steps for L-stable finite difference methods for linear second-order parabolic equations with spatially variable coefficients. The regularity properties for first and second spatial differences of the numerical solution mimic those of the continuous problem, with logarithmic factors in second differences. The regularity results for the inhomogeneous problem imply that the uniform rate of convergence of the numerical solution and its differences is controlled only by the maximum norm of the local truncation error.