Arithmetic complexity of solutions of the Dirichlet problem

The classical Dirichlet problem on the unit disk can be solved by various numerical methods. Two standard approaches are the evaluation of the Poisson integral and, via Dirichlet’s principle, the solution of an associated minimization problem. For practical applications, these methods must be realized on concrete computational platforms. In this paper, we investigate their implementation on Turing machines, the fundamental model of digital computation. We show that, in general, neither approach yields a Turing-computable solution, even when the boundary function is computable. We then provide a precise characterization of this non-computability within the Zheng-Weihrauch hierarchy and establish lower and upper bounds for the degree of non-computability associated with each approach.