Convergence of a boundary integral method for 3D interfacial Darcy flow with surface tension

We study convergence of a boundary integral method for 3D interfacial flow with surface tension when the fluid velocity is given by Darcy's Law. The method is closely related to a previous method developed and implemented by Ambrose, Siegel, and Tlupova, in which one of the main ideas is the use of an isothermal parameterization of the free surface. We prove convergence by proving consistency and stability, and the main challenge is to demonstrate energy estimates for the growth of errors. These estimates follow the general lines of estimates for continuous problems made by Ambrose and Masmoudi, in which there are good estimates available for the curvature of the free surface. To use this framework, we consider the curvature and the position of the free surface each to be evolving, rather than attempting to determine one of these from the other. We introduce a novel substitution which allows the needed estimates to close.