CVC Seminar: Wed Apr 27 1:30-3pm ACES 4.304

Andrew Gillette

Stability of Dual Discretization Methods for PDEs

We analyze the approximation of solutions to PDEs over domains discretized by the dual of a simplicial mesh. While 'primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh, a `dual discretization method’ associates DoFs with the geometric duals of these objects. Prior efforts toward such methods are unified here and motivated in a canonical way through the language of discrete exterior calculus. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties.