  We study the question of constructive approximation of the harmonic measure
$\omega_x^\Omega$ of a connected bounded domain $\Omega$ with respect to a
point $x\in\Omega$. In particular, using a new notion of computable harmonic
approximation, we show that for an arbitrary such $\Omega$, computability of
the harmonic measure $\omega^\Omega_x$ for a single point $x\in\Omega$ implies
computability of $\omega_y^\Omega$ for any $y\in \Omega$. This may require a
different algorithm for different points $y$, which leads us to the
construction of surprising natural examples of continuous functions that arise
as solutions to a Dirichlet problem, whose values can be computed at any point
but cannot be computed with the use of the same algorithm on all of their
domain. We further study the conditions under which the harmonic measure is
computable uniformly, that is by a single algorithm, and characterize them for
regular domains with computable boundaries.
