  We establish estimates on the error made by the Ritz method for quadratic
energies on the space $H^1(\Omega)$ in the approximation of the solution of
variational problems with different boundary conditions. Special attention is
paid to the case of Dirichlet boundary values which are treated with the
boundary penalty method. We consider arbitrary and in general non linear
classes $V\subseteq H^1(\Omega)$ of ansatz functions and estimate the error in
dependence of the optimisation accuracy, the approximation capabilities of the
ansatz class and - in the case of Dirichlet boundary values - the penalisation
strength $\lambda$. For non-essential boundary conditions the error of the Ritz
method decays with the same rate as the approximation rate of the ansatz
classes. For the boundary penalty method we obtain that given an approximation
rate of $r$ in $H^1(\Omega)$ and an approximation rate of $s$ in
$L^2(\partial\Omega)$ of the ansatz classes, the optimal decay rate of the
estimated error is $\min(s/2, r) \in [r/2, r]$ and achieved by choosing
$\lambda_n\sim n^{s}$. We discuss how this rate can be improved, the relation
to existing estimates for finite element functions as well as the implications
for ansatz classes which are given through ReLU networks. Finally, we use the
notion of $\Gamma$-convergence to show that the Ritz method converges for a
wide class of energies including nonlinear stationary PDEs like the
$p$-Laplace.
