  In this paper, we study the Dirichlet problem for Laplace's equation in an
open disk. The uniqueness of solutions is ensured by the well-known weak
maximum principle. We introduce a novel approach to demonstrate the existence
of a solution using harmonic polynomials that converge uniformly to a solution.
Specifically, we rigorously derive the convergence rate of the harmonic
polynomials and show that smoother boundary data and proximity of the target
point to the disk's origin accelerate the convergence. Additionally, we obtain
uniform estimates for the derivatives of solutions of arbitrary orders,
controlled by $L^1$-boundary data. Notably, the constants in our estimates are
significantly improved compared to existing results. Furthermore, we provide an
enhanced radius of convergence for Taylor's series of the solution at each
point in the open disk.
