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.

翻译：单位圆盘上的经典狄利克雷问题可通过多种数值方法求解。两种最常用且普遍采用的方法是：对关联的泊松积分进行积分，以及通过应用狄利克雷原理求解特定最小化问题。在实际应用中，这些流程需在具体计算平台上实现。本文研究这些流程在图灵机（任何数字计算机的基本模型）上的实现。我们证明，在此计算平台上，即使边界函数是可计算的，求解狄利克雷问题的两种方法通常仍会产生非图灵可计算的解。随后，本文通过郑-魏劳赫分层对此不可计算性进行了精确刻画。针对两种方法，我们推导了其在郑-魏劳赫分层中不可计算度的下界与上界。