Reproducing utility of positive definiteness for Stokes flow with AMR

Many important physical problems require numerical methods that use a range of solution resolution scales, usually realized through the use of adaptive mesh refinement (AMR). The introduction of AMR into a solver can result in lower accuracy or a loss of symmetric positive definiteness (SPD). This is especially relevant in fluctuating hydrodynamics, as the structure of the Laplace operator is exploited in satisfying the fluctuation-dissipation-balance by setting the noise porportional to the square root of the viscous operator. This work describes two second order accurate methods for constructing a Laplace operator using the marker and cell (MAC) method for Stokes flow with adaptive mesh refinement that reproduce the utility of the SPD operator for uniform meshes. In one approach, Galerkin differencing is used to force the resulting system to by SPD, yielding a result with a nontrivial square root. And in the other, the resulting operator is not SPD, but is constructed in a way that defines a matrix that can be used to satisfy the fluctuation-dissipation-balance.