An Introduction to the Theory of M-Decompositions
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.
The authors would like to express their gratitude to Daniele Antonio Di Pietro, Alexander Ern and Luca Formaggia for their invitation to write this paper. The first author would thank them for the invitation to give a couple of lectures on HDG methods as part of the Introductory School (to the IHP quarter on Numerical Methods for PDEs) they organized in September 5–9, 2016, at the Institut d’Études Scientifiques de Cargèse, in Corse, France. Part of the material of those lectures is further developed here.
The author “Bernardo Cockburn” was partially supported by National Science Foundation grant DMS 1522657.
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