| Description |
This work is based on the work of Milton, Seppecher, and Bouchitte on variational principles for lossy media. I will describe several applications of the variational principle for the complex Helmholtz equation. First, I will describe a nite element method for solving the Helmholtz equation based on the minimization variational principle. The matrix in the linear system that results from the nite element method is symmetric positive defnite, due to the minimization variational principle upon which it is based. I also present an error bound for the nite element method and an effective preconditioning strategy that can be used when the linear system is solved with the preconditioned conjugate gradient method. Another application is the extension of the variational principles to handle more general boundary conditions, such as Robin boundary conditions. We base the Robin formulation on the natural boundary conditions. The importance of the Robin condition is that it can be thought of as a first order approximation to a transmission boundary condition, which is useful for scattering problems. Next, I use the minimization variational principle to formulate a method for the tomography problem for Helmholtz equation. The tomography problem is to determine the coffecients of the Helmholtz equation from knowledge of the essential to natural map. The basic idea behind this method is to minimize the L2 distance between the solutions to the essential and natural problems for a given set of essential and natural data. However, instead of solving the natural and essential problems, we add the functionals for which the solutions are minimizers as constraints in the minimization problem outlined above. Then all the parameters in the minimization (solutions and material coffecients) are considered as independent. Then the minimization is performed with respect to all the parameters. A regularization term on the material coeffcients is added to stabilize the solution. The final application will be to provide some elementary bounds on the essential to natural map by applying the variational principles with simple test functions. These bounds can be especially useful in determining volume fraction information from the measurements of current and potential on the boundary. |
