Colloquium

Time-harmonic acoustic wave propagation problems modeled by the Helmholtz equation have many applications. We consider wave propagation in car cabins, scattering by targets in sediment, and geological surveys. In these applications, the diameter of computational domains is often tens or hundreds of wavelengths. We consider the accuracy of discretizations and show that a rule of thump to have about 10 unknowns per wavelength is often reasonable. Thus, the above applications lead to discrete problems with millions or billions of unknowns. These huge problems require efficient solution methods. We consider an iterative method with a damped preconditioner employing a multigrid method. Currently this is a popular approach. We also briefly mention some other solution methods we have studied and show their performance on our example problems.