Chapter 2 An Introduction to the Theory of M -Decompositions Bernardo Cockburn, Guosheng Fu, and Ke Shi Abstract We provide a short introduction to the theory of M -decompositions in the framework of steady-state diffusion problems. This theory allows us to systematically devise hybridizable discontinuous Galerkin and mixed methods which can be proven to be superconvergent on unstructured meshes made of elements of a variety of shapes. The main feature of this approach is that it reduces such an effort to the definition, for each element K of the mesh, of the spaces for the flux, V (K), and the scalar variable, W(K), which, roughly speaking, can be decomposed into suitably chosen orthogonal subspaces related to the space traces on ∂ K of the scalar unknown, M(∂ K). We begin by showing how a simple a priori error analysis motivates the notion of an M -decomposition. We then study the main properties of the M -decompositions and show how to actually construct them. Finally, we provide many examples in the two-dimensional setting. We end by briefly commenting on several extensions including to other equations like the wave equation, the equations of linear elasticity, and the equations of incompressible fluid flow. 2.1 Introduction The theory of M -decompositions has been recently introduced as an effective tool to systematically find the local spaces defining hybridizable discontinuous Galerkin and mixed methods which can be proven to be superconvergent on unstructured B. Cockburn () School of Mathematics, University of Minnesota, Minneapolis, MN, USA e-mail: cockburn@math.umn.edu G. Fu Division of Applied Mathematics, Brown University, Providence, RI, USA e-mail: guosheng_fu@brown.edu K. Shi Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA, USA e-mail: kshi@odu.edu © Springer Nature Switzerland AG 2018 D. A. Di Pietro et al. (eds.), Numerical Methods for PDEs, SEMA SIMAI Springer Series 15, https://doi.org/10.1007/978-3-319-94676-4_2 5 cockburn@math.umn.edu