  We estimate the error of the Deep Ritz Method for linear elliptic equations.
For Dirichlet boundary conditions, we estimate the error when the boundary
values are imposed through the boundary penalty method. Our results apply to
arbitrary sets of ansatz functions and estimate the error in dependence of the
optimization accuracy, the approximation capabilities of the ansatz class and
-- in the case of Dirichlet boundary values -- the penalization strength
$\lambda$. To the best of our knowledge, our results are presently the only
ones in the literature that treat the case of Dirichlet boundary conditions in
full generality, i.e., without a lower order term that leads to coercivity on
all of $H^1(\Omega)$. Further, we discuss the implications of our results for
ansatz classes which are given through ReLU networks and the relation to
existing estimates for finite element functions. For high dimensional problems
our results show that the favourable approximation capabilities of neural
networks for smooth functions are inherited by the Deep Ritz Method.
