respectively Dirichlet and Neuman boundary conditions for the tangential components of the metric. For the Einstein-DeTurck equation, we have an additional $D$ equations and therefore need an additional $D$ boundary conditions. Since we want to obtain a solution with $\xi^\mu=0$, and since is strictly elliptic, it seems clear that we should require $\xi^\mu|_{\partial M} = 0$. If we denote the normal and tangential components of $\xi^\mu$ by $\xi^n$ and $\xi^i$ respectively, then it is easy to show that $\xi^n|_{\partial M} = 0$ fixes the normal derivative of $g_{nn}$, while $\xi^i|_{\partial M} = 0$ fixes the normal derivative of $g_{ni}$, in terms of their values, and the values and derivatives of the other metric components. (This is the same issue as imposing boundary conditions for the parabolic PDE mentioned above, $\partial g_{\mu\nu}/\partial\lambda = -2R^H_{\mu\nu}$. In [@Headrick:2006ti] it was argued that $\xi^\mu|_{\partial M} = 0$ are appropriate boundary conditions for that PDE, and uniqueness has been proved in that context [@Holzegel:2007zz].)

Algorithms for solving the Einstein-DeTurck equation {#sec:algorithms}
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The Einstein-DeTurck equation is a strictly elliptic nonlinear PDE, and there exist many standard methods for solving such equations. As mentioned above, most of these are relaxation methods, meaning that one starts with 