Douglas Arnold's research interests include numerical analysis, partial differential equations, classical and quantum mechanics, mathematical physics, and in particular, the interplay between these fields. Much of his work concerns the computer solution of partial differential equations, focusing on the development and understanding of methods for simulating physical phenomena ranging from the deformation of elastic plates and shells to the collision of black holes. Around 2002 he initiated the finite element exterior calculus, a new approach to the stability of finite element methods based on geometric and topological structure underlying the relevant partial differential equations. The development and applications of the finite element exterior calculus forms one of the two major directions of his current research work. The other is the study of wave localization in disordered media, a fascinating topic with numerous important applications, for example, to the design of more efficient, and so more sustainable, LED lighting.