We develop spatially adaptive numerical methods for two distinct stochastic physical systems. The first is a model for the kinetics of supported lipid bilayer formation by the adsorption and rupture of lipid vesicles onto a solid substrate. We model the adsorption process taking into account the distinct vesicle rupture events and growth processes. This includes (i) the initial adhesion and vesicle rupture that nucleates bilayer islands, (ii) the growth and merger of bilayer islands, (iii) enhanced adhesion of vesicles to the bilayer edge, and (iv) the final desorption of excess vesicles from the substrate. These simulation studies give insight into prior experimental observations of adsorption in which an overloading of lipid on the solid substrate occurs before formation of the final supported lipid bilayer. Our model provides an explanation for the features of the interesting universal master curve that was observed for the surface fluorescence intensity in the experimental investigations of Weirich et. al. The second method describes a model for fluid-structure interactions when subject to thermal fluctuations through a mixed finite element discretization. These are developed for incompressible Stokes flow with properties that facilitate the introduction of stochastic driving fields to account for thermal fluctuations in a manner consistent with statistical mechanics. To generate the stochastic driving fields efficiently, we develop stochastic iterative methods having O(N) computational complexity. Our approach provides an alternative to uniform discretizations on periodic domains and FFTs. Our stochastic numerical methods allow for spatially adaptive simulations of fluid-structure interactions subject to thermal fluctuations in the presence of walls with no-slip boundary conditions and on domains having complex geometries.