Spectral Penalty Methods

Jan Hesthaven
Division of Applied Mathematics
Brown University

Abstract: One of the central problems in developing and analyzing high-order methods, and in particular spectral methods, for the solution of partial differential equations results from the need to impose boundary conditions. In the traditional approach boundary conditions are enforced strongly by directly modifying the operators. This approach, however, is well known to become increasingly complicated for complex boundary operators and it significantly complicates the analysis of the complete scheme.

A technique for overcoming some, if not all, of the aforementioned problems is offered by the spectral penalty methods in which one enforces the boundary conditions in a weak way only, albeit to the order of the numerical scheme.

In this talk we shall review the development and formulation of the spectral penalty methods which, as shall be shown through a number of examples, alleviates many of the above problems and provides a constructive approach for the development of asymptotic stable schemes for the solutions of partial differential equations. Moreover, the separation of the operator and the boundary conditions turns out to be very fruitful in terms of facilitating analysis.

To illustrate the analysis as well as the versatility of the penalty methods we shall discuss the construction of stable schemes for problems with non-trivial boundary conditions, the formulation of very efficient preconditioners for pseudospectral operators and the development of stable spectral schemes for the solution of partial differential equations on generally unstructured grids in two and three dimensions. Examples of applications from gasdynamics and electromagnetics will illustrate the success of the proposed schemes.