Arithmetic Complexity of Solutions of the Dirichlet Problem

TL;DR

This paper investigates the non-computability of classical solutions to the Dirichlet problem on the unit disk when implemented on Turing machines, providing a detailed hierarchy-based characterization of their computational complexity.

Contribution

It demonstrates that standard numerical approaches to the Dirichlet problem are generally non-Turing computable and characterizes this non-computability within the Zheng--Weihrauch hierarchy.

Findings

Solutions are not Turing computable even with computable boundary functions.

Provides bounds on the degree of non-computability in the Zheng--Weihrauch hierarchy.

Characterizes non-computability for both Poisson integral and Dirichlet's principle approaches.

Abstract

The classical Dirichlet problem on the unit disk can be solved by different numerical approaches. The two most common and popular approaches are the integration of the associated Poisson integral and, by applying Dirichlet's principle, solving a particular minimization problem. For practical use, these procedures need to be implemented on concrete computing platforms. This paper studies the realization of these procedures on Turing machines, the fundamental model for any digital computer. We show that on this computing platform both approaches to solve Dirichlet's problem yield generally a solution that is not Turing computable, even if the boundary function is computable. Then the paper provides a precise characterization of this non-computability in terms of the Zheng--Weihrauch hierarchy. For both approaches, we derive a lower and an upper bound on the degree of non-computability in the Zheng--Weihrauch hierarchy.