Search

We introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities. For this purpose, we use an output least squares approach with level set and augmented Lagrangian methods. Our formulation incorporates the inherent nature of the piecewise constant coefficients, which eliminates the need for a complicated nonlinear solve at every iteration. Instead, we obtain an explicit update formula and therefore vastly speed up computation. We employ our approach to the example problems of Poisson's equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface.

The immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid. The immersed elastic material is represented in the fluid equations by a system or field of applied forces. The particular case of Stokes flow with applied forces on a periodic domain involves two related mathematical complications. One of these is that an arbitrary constant vector may be added to the fluid velocity, and the other is the constraint that the integral of the applied force must be zero. Typically, forces defined on a freely floating elastic immersed boundary or body satisfy this constraint, but there are many important classes of forces that do not. For example, the so-called tether forces that are used to prescribe the simulated configuration of an immersed boundary, possibly in a time-dependent manner, typically do not sum to zero. Another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system. We present a method for periodic Stokes flow that when used with tether points, admits the use of all forces irrespective of their integral over the domain. A byproduct of this method is that the additive constant velocity associated with periodic Stokes flow is uniquely determined. Indeed, the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied.