Adaptive mesh refinement for pseudospectral methods in numerical relativity Across all of computational physics a central problem is that of discretization from the choice of resolution in simple finite difference approaches to the details of more intricate discretization schemes such as spectral elements. The choice of discretization decides the numerical solution space as well as the properties of numerical methods such as their convergence and stability. For this reason the effective use of any numerical scheme requires a proper understanding of the underlying discretization scheme and its parameters. In particular modern numerical methods often incorporate adaptive discretization schemes utilizing heterogeneous meshes that change with time. In this work we will explore one such method in the form of a state of the art numerical relativity code and the implementation of an adaptive mesh refinement (AMR) scheme within it. We describe in detail its features and the resulting properties as it is used to solve physical problems in the form of hyperbolic partial differential equations and we examine the scaling behavior of the resulting method. We also present results obtained using this scheme in the form of simulations of the critical collapse of gravitational waves that were made possible by the AMR system showing some evidence of both self similarity and universality in this system. Finally we study a suite of several challenging test cases beginning with a simple two dimensional wave equation with an added nonlinearity which results in critical behavior for certain choices of initial data then moving on to the collapse of a real scalar field minimally coupled to general relativity in spherical symmetry. Finally we use the collapse of gravitational waves in vacuum in axisymmetry as our third test case. We use these example problems to evaluate the gains in terms of accuracy as well as efficiency that are obtained through the use of adaptive resolutions.