Arithmetic Complexity of Solutions of the Dirichlet Problem

2026-01-11

arXiv.org

AI Summary

This study investigates the computability of numerical solutions to the Dirichlet problem on the unit disk within the Turing machine model. Focusing on two classical solution approaches—Poisson integration and the Dirichlet principle—it provides the first rigorous characterization of the degree of non-computability of their solutions by integrating tools from computable analysis and the Zheng–Weihrauch hierarchy. The results demonstrate that even when the boundary function is computable, the resulting solution is generally not Turing-computable. Moreover, the work establishes precise upper and lower bounds for each method within the Zheng–Weihrauch hierarchy, thereby revealing their intrinsic arithmetical complexity and filling a significant gap in the computability-theoretic understanding of this classical problem.

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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.

Problem

• Dirichlet problem
• Turing computability
• Poisson integral
• Zheng–Weihrauch hierarchy
• numerical methods

Innovation

• Turing computability
• Dirichlet problem
• Zheng–Weihrauch hierarchy
• Poisson integral
• computable analysis