Adaptive meshing is important to many practical problems, though it presents many challenges. This work focuses primarily on mesh refinement within the context of Mimetic Spectral Element Method (MSEM). This is a method that approaches formulating the discrete system of equations in the style of Finite Element Method, but through application of differential geometry and exterior calculus. This results in the method satisfying the inf-sup condition by construction, allowing it to correctly solve any saddle point problem, such as the incompressible Navier-Stokes equations. This work also examines the applicability of using results given by Variational Multi-Scale (VMS) theory as an error estimator. The solver that was written based on the theory in this work and then later used to produce the results can be found at the remote repository under the GNU Public License 3.

MSEM in this work is formulated in a hybridized way, where each element is considered to have separate degrees of freedom, with continuity being enforced through Lagrange multipliers. This allows for neighboring elements to have different polynomial orders and even levels of refinement. Local refinement is achieved using raising polynomial order of elements (p-refinement) and hierarchically dividing elements (h-refinement). To this end, the theory and procedures to implement combined hp-refinement within the existing MSEM framework are formulated and verified.

From there, a refinement criterion for deciding between h-refinement or p-refinement for each element involved in a refinement sweep is introduced and validated. This refinement criterion is based on estimating the change in the L² error norm of the solution and is computed by using the Legendre coefficients of the error estimate for each element. This is in contrast to many other similar criteria present in literature, which consider only the solution and the current mesh state as the basis of the decision to apply either h-refinement or p-refinement.

To obtain an error estimate to use for the refinement criterion, several different estimators are proposed, notably including VMS. The main appeal of using VMS is that a good error estimate is offered by the unresolved scales, which are obtained by the method normally. These error estimates were tested on steady, two-dimensional test problems of increasing complexity, from mixed formulation Poisson equation, to linear advection-diffusion, and lastly incompressible Navier-Stokes equations. Based on these tests, VMS appears second only to knowing the exact error, though the computational cost associated with each refinement criterion was not compared.