Convergence in the maximum norm of ADI-type methods for parabolic problems

TL;DR

This paper establishes unconditional convergence in the maximum norm for ADI-type methods applied to semilinear parabolic problems with multiple spatial dimensions, under general conditions and boundary constraints.

Contribution

It provides the first general proof of maximum norm convergence for ADI methods in higher dimensions with Dirichlet boundary conditions.

Findings

Proves power-boundedness of the stability function independently of space and time resolutions.

Establishes unconditional convergence in the maximum norm for ADI methods in multiple dimensions.

Applies to a broad class of semilinear parabolic problems.

Abstract

Results on unconditional convergence in the Maximum norm for ADI-type methods, such as the Douglas method, applied to the time integration of semilinear parabolic problems are quite difficult to get, mainly when the number of space dimensions $m$ is greater than two. Such a result is obtained here under quite general conditions on the PDE problem in case that time-independent Dirichlet boundary conditions are imposed. To get these bounds, a theorem that guarantees, in some sense, power-boundeness of the stability function independently of both the space and time resolutions is proved.