TY - CHAP

T1 - An Introduction to the Theory of M-Decompositions

AU - Cockburn, Bernardo

AU - Fu, Guosheng

AU - Shi, Ke

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85055031463&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055031463&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-94676-4_2

DO - 10.1007/978-3-319-94676-4_2

M3 - Chapter

AN - SCOPUS:85055031463

T3 - SEMA SIMAI Springer Series

SP - 5

EP - 29

BT - SEMA SIMAI Springer Series

PB - Springer International Publishing

ER -