Colloquium - 9/16, 3:00PM, Duncan Hall 1064

Jesse Chan

Department of Computational and Applied Mathematics
Rice University

"Dual Petrov-Galerkin Methods: An Overview of Theory and Applications"

The finite element method is often referred to as the method of weighted residuals, where the solution is recovered by enforcing that the residual is zero when weighted by specific weighting (test) functions. Petrov-Galerkin finite element methods recognize that, for certain problems, the choice of such test functions greatly impacts the efficiency and stability of the method. We introduce the concept of Dual Petrov-Galerkin methods, a class of minimum residual finite element methods in which the finite element residual is minimized over the dual norm on the space of test functions. Under this functional setting, this method can be re-interpreted as a Petrov-Galerkin finite element method, where the weighting functions are tailored to respect the underlying structure of the physics and discretization.

We derive two families of methods that fall under the framework of dual Petrov-Galerkin methods. We begin with the Discontinuous Petrov-Galerkin method, a hybridized discontinuous Galerkin scheme with optimal test functions computed on-the-fly for each individual problem. Numerical examples of are given in 2D for singular perturbation problems such as Helmholtz and convection-diffusion, as well as for other classically stability-challenged problems such as Stokes flow and electromagnetics. The method is described for nonlinear problems as well, with examples from incompressible/compressible Navier-Stokes and a simple seismic inversion problem. We then extend the Dual Petrov-Galerkin methodology to arbitrary discretizations using a saddle point framework, and discuss preliminary results for convection-diffusion using continuous finite elements.