Multiscale Finite Elements for Acoustics: Continuous, Discontinuous, and Stabilized Methods

This work describes two perspectives for understanding the numerical difficulties that arise in the solution of wave problems, and various advances in the development of efficient discretization schemes for acoustics. Standard, low-order, continuous Galerkin finite element methods are unable to cope with wave phenomena at short wave lengths because the computational effort required to resolve the waves and control numerical dispersion errors becomes prohibitive. The failure to adequately represent subgrid scales misses not only the fine-scale part of the solution, but often causes severe pollution of the solution on the resolved scale as well. Since computation naturally separates the scales of a problem according to the mesh size, multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them. The Galerkin/least squares method arises in multiscale settings, and its stability parameter is defined by dispersion considerations. Bubble enriched methods employ auxiliary functions that are usually expressed in the form of infinite series. Dispersion analysis provides guidelines for the implementation of the series representation in practice. In the discontinuous enrichment method, the fine scales are spanned by free-space homogeneous solutions of the governing equations. These auxiliary functions may be discontinuous across element boundaries, and continuity is enforced weakly by Lagrange multipliers.