This work concerns numerical methods for material interfaces moving in viscous fluid flow. Such models are used especially for biological processes on small scales, in which the interface exerts a force on the fluid. The work to be done will focus on the efficient computation of surface integrals representing viscosity-dominated flow, or Stokes flow, and error analysis of finite difference methods for more general Navier-Stokes flow, such as the immersed interface method, in which the equations are discretized on a regular grid, with modifications near the interface to account for the forces. The methods considered should be at least second-order accurate. In the first project a relatively simple method will be developed for computing singular or nearly singular integrals on smooth surfaces, such as the velocity integrals for three-dimensional Stokes flow, evaluated on or near the surface. This work will improve and generalize earlier work, in which a standard quadrature of a regularized integral is combined with corrections found by analysis near the singularity. This method should be accurate even for grid points near the surface, allowing more flexibility in computing the motion of the surface. It could be used as part of a computation for Navier-Stokes flow. In the second project, error estimates in maximum norm will be derived for finite difference methods such as the immersed interface method. Recent results of the proposer showing a gain in regularity for finite difference versions of Poisson or diffusion equations will be used to clarify the relationship between the accuracy of numerical solutions and the corrections needed near the interface and also the choice of time discretization. The work will include convergence proofs for simplified interface problems with Navier-Stokes flow and maximum norm stability of the approximate projection on divergence-free vector fields.
A number of scientific problems involve moving boundaries in fluids, such as a drop of one fluid in another, or the motion of an elastic membrane in living tissue. Numerical study of such problems has special difficulties, since it is desirable to discretize the fluid variables on a fixed grid, while the moving boundary must be represented separately, together with its influence on the fluid motion. It is difficult to discretize the moving surface in a way that is simple and practical. For Stokes flow, dominated by viscosity, integral formulations of the fluid variables are widely used. The proposed method of integration promises to be simpler and more efficient than standard methods and require less effort with the surface. Thus it could contribute to the practicality of three-dimensional simulations as the models become more realistic. The second project emphasizes estimates of maximum errors, since these are likely to be largest near interfaces. They are less well developed than estimates in integral norms. Such estimates will be used for methods such as the immersed interface method, the decomposition of Navier-Stokes flow mentioned above, and approximate projection methods. Error analysis of existing numerical methods should improve understanding of their validity and limitations. It can also show how choices in the methods affect their accuracy and suggest improvements.
