Xiaoming ZhengCentral Michigan University (Host Kun Zhao)
This talk presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by the projection of mesh nodes onto the interface and the insertion right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. This is a joint work with John Lowengrub of University of California at Irvine.
Cecilia MondainiTexas A&M University (Host Vincent Martinez)
We consider a data assimilation algorithm for recovering the exact value of a reference solution of the two-dimensional Navier-Stokes equations, by using continuous in time and coarse spatial observations. The algorithm is given by an approximate model which incorporates the observations through a feedback control (nudging) term. We obtain an analytical uniform in time estimate of the error committed when numerically solving this approximate model by using a post-processing technique for the spectral Galerkin method, inspired by the theory of approximate inertial manifolds. This is a joint work with C. Foias and E. S. Titi.
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