Meshless methods for the detection of an obstacle submerged in a two-dimensional Stokes flow

We consider a geometric inverse problem which requires detecting an unknown obstacle, e.g. a submarine or an aquatic mine, submerged in a Stokes slow viscous stationary flow of an incompressible fluid. In two-dimensions, the problem is formulated in terms of the biharmonic streamfunction in an unbounded domain which is approximated using the Trefftz method and the method of fundamental solutions (MFS). This is, apparently, the first time the Trefftz method and the MFS are applied for the solution of the biharmonic equation in an unbounded domain. We first examine direct problems and then consider inverse problems. The unknown obstacle is determined by employing a nonlinear Tikhonov regularization procedure. Numerical results are presented and discussed.