2007 PIMS-CSC Seminar - 02

Wave propagation phenomena are present and explored in a variety of applications, ranging from medical imaging to radar detection. Those sophisticated applications require accurate modeling in order to optimize the involved procedures. For that, classical analytical techniques seems to be not viable, what leads to the need of resorting to numerical methods.

As a first step towards developing reliable numerical models of wave propagation, the Helmholtz equation, which governs time-harmonic acoustic, elastic and electromagnetic waves, is to be tackled. Numerical approximation of this equation is particularly challenging as reported in a vast literature. The oscillatory behavior of the exact solution and the quality of the numerical approximation depend on the wave number $k$. To approximate Helmholtz equation with
acceptable accuracy the resolution of the mesh should be adjusted to the wave number according to a rule of thumb , which prescribes a minimum number of elements per wavelength. It is well known that, despite of the adoption of this rule, the performance of the Galerkin finite element method deteriorates as $k$ increases. This misbehavior, known as pollution of the finite element solution , can only be avoided after a drastic refinement of the mesh, which normally entails significant barriers for the numerical analysis of Helmholtz equation at mid and high frequencies.

A great effort has been devoted to alleviate pollution effects . In particular, the GLS method (Galerkin Least-Squares) is able to completely eliminate the phase lag in one dimension problems . Nevertheless, in two-dimensions this method is not pollution-free for any direction of a plane wave . In fact, in two space
dimensions, there is no finite element method with piecewise linear shape functions free of pollution effect. Stencils with minimal pollution error are constructed in through the Quasi Stabilized Finite Element Method (QSFEM). QSFEM is really a finite difference rather than a finite element method, as the modifications of the discrete operator are made on the algebraic level and no variational formulation is associated with its original form.

In the present talk, some enhanced Finite Element Formulations recently developed by the author and collaborators are discussed. Those rely on either discontinuous interpolations or projections of residual values of balance equations. The talk also addresses the important issue of uncertainty propagation when Finite Element formulations are applied within numerical simulation of acoustic problems.