This thesis aims to introduce mesh refinement into the Mimetic Spectral Element Method (MSEM). The concept of mimetic discretizations is to mimic the properties of continuous Partial Differential Equations (PDEs) discretely. In many discretization methods information is lost in the actual discretization step which is detrimental to the physical fidelity of the approximated solution. Mimetic methods try to prevent this, a feat achieved by combining the fields of differential geometry and algebraic topology. Where differential geometry describes the continuous problem, algebraic topology functions as its discrete equivalent. By accounting for the spatial and temporal geometric objects each physical quantity is associated with, mimetic methods preserve as much as possible of the continuous structure.