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Advanced Numerical Methods in the Mathematical Sciences

Jay Gopalakrishnan, Portland State University
Traces of a Friedrichs Space for the Wave Equation and Tent Pitching


The time-dependent wave equation can be regarded as a Friedrichs system. The wave operator then becomes a continuous operator on a Hilbert space normed with a graph norm. In this talk, focusing on the case of one space dimension, we discuss a trace theory for this space. In order to set up a weak formulation using the modern theory of Friedrichs systems, one must incorporate the boundary conditions into an intrinsic double cone within this space. This is done using the trace theory. Of particular interest are domains whose inflow and outflow boundaries intersect. Such domains arise in explicit finite element schemes of the tent pitching type. We find that the traces have a weak continuity property at the meeting points of the inflow and outflow boundaries. Motivated by this continuity property, we use discrete spaces that conform to the continuity property, thus resulting in a new explicit scheme with locally variable time and space mesh sizes.