With the rise of computational power and the democratized access to supercomputers, numerical simulations have emerged as a new standard tool of scientific investigation over the last century, often complementing experimental and theoretical approaches. Among the most challenging problems to be tackled numerically, one finds multi-scale and free-boundary phenomena. Multi-scale problems typically prevent the use of uniform meshes; free boundaries may develop complex geometric features making body-fitted mesh generation challenging. The present dissertation covers the development of numerical tools and methods for implicitly-captured interfaces on distributed Quadtree/Octree grids, which are thus well-suited for such challenging problems.

The first part of this work focuses on the extension of numerical methods for the simulation of incompressible, viscous flows from single-phase to two-phase problems, on distributed Quadtree/Octree grids. In order to reconcile the numerical description of the phenomenon with its continuum-mechanics description, this extension requires the development of several sharp-interface numerical techniques capable of capturing discontinuities in material parameters as well as in the primary unknowns. In addition, the description of the first-principle conservation laws across a sharp interface underlines that interface discontinuities in primary unknowns depend on the solution itself, highlighting the need for novel numerical methods. The development of such methods is presented, and their combination into a simulation engine for incompressible, viscous two-phase flows is illustrated.

In the second part of this dissertation, two separate and independent works are presented. First, a numerical discretization for the point-located Dirac distribution is presented. Beyond its theoretical interest, the Dirac distribution is sometimes used for modeling extreme multiscale events that cannot be fully resolved even with adaptive grid capabilities: that project presents a simple, geometric discretization that is shown to correctly reproduce the expected behavior over finite length scales. Second, a series of algorithms implementing a parallel, load-balanced divide-and-conquer strategy for constructing levelset representations of the Solvent-Excluded Surfaces of large biomolecular compounds are presented. This set of algorithms is shown to successfully accelerate the construction of such highly convoluted interfaces and to satisfy strong scaling, opening the door to more efficient calculations of interface-resolved electrostatics phenomena around such complex molecular structures.