Maximum Norm Regularity of Implicit Difference Methods for Parabolic Equations

We prove maximum norm regularity properties of L-stable finite difference

methods for linear-second order parabolic equations with coefficients

independent of time, valid for large time steps. These results are almost

sharp; the regularity property for first differences of the numerical solution

is of the same form as that of the continuous problem, and the regularity

property for second differences is the same as the continuous problem except for

logarithmic factors.

This generalizes a result proved by Beale valid for the constant-coefficient

diffusion equation, and is in the spirit of work by Aronson, Widlund and

Thomeé.

To prove maximum norm regularity properties for the homogeneous problem,

we introduce a semi-discrete problem (discrete in space, continuous in time).

We estimate the semi-discrete evolution operator and its spatial differences on

a sector of the complex plan by constructing a fundamental solution.

The semi-discrete fundamental solution is obtained from the fundamental solution to the frozen coefficient problem by adding a correction term found through an iterative process.

From the bounds obtained on the evolution operator and its spatial differences,

we find bounds

on the resolvent of the discrete elliptic operator and its differences through

the Laplace transform

representation of the resolvent. Using the resolvent estimates and the

assumed stability properties of the time-stepping method in the Cauchy integral

representation of the fully discrete solution operator

yields the homogeneous regularity result.

Maximum norm regularity results for the inhomogeneous

problem follow from the homogeneous results using Duhamel's principle. The results for the inhomogeneous

problem

imply that when the time step is taken proportional to the grid width, the rate of convergence of the numerical solution and its first

differences is second-order in space, and the rate of convergence for second

differences

is second-order except for logarithmic factors .

As an application of the theory, we prove almost sharp maximum norm resolvent estimates for divergence

form elliptic operators on spatially periodic grid functions. Such operators are invertible, with inverses and their first differences bounded in maximum norm, uniformly in the grid width. Second differences of the inverse operator are bounded except for logarithmic factors.