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Mathematics Colloquium: Fictitious Domain Methods for Time-Dependent Wave Propagation Problems
4:10 pm, Neill Hall 5W
Abstract: A fictitious domain method (FDM) is a technique in which the solution to a given problem is obtained by extending the given data to a larger but simpler shaped domain, containing the original domain, and solving corresponding equations in this fictitious domain. FDMs have mostly been used for solving problems in the stationary case. FDMs for time dependent problems are relatively new and have been studied by R. Glowinski and P. Joly among others. One class of fictitious domain methods involves using distributed/boundary Lagrange multipliers to enforce the boundary conditions on the original smaller domain. This is known as the functional analytic approach and leads to saddle point problems. In this talk I will describe a distributed multiplier FDM for the model problem of propagation of waves in an exterior domain, i.e., the (unbounded) complement of a bounded obstacle. For such problems, the FDM is also called the domain embedding method. As the problems considered are defined on unbounded domains, I will also discuss the use of perfectly matched layers (PMLs) to handle the truncation of the domain efficiently. In addition, I will describe an operator splitting method in conjunction with this distributed multiplier FDM for the model problem.