EU Regional School - Bochev Seminar
Prof. Dr. Pavel Bochev - High-Order Methods in Numerical Simulation
Politecnico di Torino
Variational principles take advantage of the intrinsic connections between the structure of many PDE and optimization problems to identify compatible discretizations. Differential complexes provide another tool to encode the structure of a PDE. Differential forms represent global quantities rather than fields, and provide a model for the way we observe physical processes. The idea that differential forms can and should be used to develop compatible (mimetic) discretizations started to permeate the computational sciences approximately two decades ago and led to fundamental advances in computational electromagnetics. Since then, geometrical approaches to discretization have enjoyed a steady and ever increasing interest and appreciation in the computational sciences. The goal of these lectures is two-fold. First we explain how variational and geometric techniques can complement each other in the quest for accurate and stable discretizations by providing tools for the design and analysis of compatible models. In doing so, we will retrace the key steps that have led to our modern understanding of these connections. Then we discuss the options at our disposal when, for whatever reason, compatible discretizations are unfeasible.