Numerical exterior algebra and the compound matrix method (Q1849402)

The authors reformulate the compound matrix method in a coordinate-free way using exterior algebra spaces \(\bigwedge^k({\mathbb C}^n)\). This formulation leads to a general framework for studying systems of differential equations on \(k\)-dimensional subspaces. The framework requires the development of new ideas, namely, the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equation on \(\bigwedge^k({\mathbb C}^n)\), a general formulation \({\mathbb C}^n)\) of \(k\)-dimesnional subspaces, and a formulation for induced systems on an unbounded interval. The numerical exterior algebra framework is most advantageous for the numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations. The theory is applied to the stability of four differential eigenvalue problems on unbounded intervals: (1) boundary-layer flow past a compliant surface, (2) solitary waves, (3) Bickley jet in oceanography, and (4) Ekman layer in atmospheric dynamics.