We present an algorithm for treating mesh refinement interfaces in numericalrelativity. We detail the behavior of the solution near such interfaces locatedin the strong field regions of dynamical black hole spacetimes, with particularattention to the convergence properties of the simulations. In our applicationsof this technique to the evolution of puncture initial data with vanishingshift, we demonstrate that it is possible to simultaneously maintain secondorder convergence near the puncture and extend the outer boundary beyond 100M,thereby approaching the asymptotically flat region in which boundary conditionproblems are less difficult and wave extraction is meaningful.
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