It is well known that semi-discrete high order discontinuous
Galerkin (DG) methods satisfy cell entropy inequalities for
the square entropy for both scalar conservation laws (\RefOne{Jiang and Shu (1994)} \cite{jiang1994cell}) and symmetric hyperbolic systems (\RefOne{Hou and Liu (2007)} \cite{hou2007solutions}), in any space dimension and
for any triangulations.
However, this property holds only for the square entropy and
the integrations in the DG methods must be exact.
It is significantly more difficult to design DG methods
to satisfy entropy inequalities for a non-square convex entropy,
and / or when the integration is approximated by a numerical
quadrature.
In this paper, we develop a unified framework for designing
high order DG methods which will satisfy entropy inequalities
for any given single convex entropy, through suitable
numerical quadrature which is specific to this given entropy.
Our framework applies from one-dimensional scalar cases all
the way to multi-dimensional systems of conservation laws.
For the one-dimensional case, our numerical quadrature is
based on the methodology established in
\RefOne{Carpenter et al (2014)} \cite{carpenter2014entropy} and \RefOne{Gassner (2013)} \cite{gassner2013skew}.
The main ingredients are summation-by-parts (SBP) operators
derived from Legendre Gauss-Lobatto quadrature, the \RefThree{entropy
conservative flux} within elements, and the entropy stable flux at
element interfaces. We then generalize the scheme to
two-dimensional triangular meshes by constructing SBP
operators on triangles based on a special quadrature rule.
A local discontinuous Galerkin (LDG) type treatment is also
incorporated to achieve the generalization to
convection-diffusion equations. Extensive numerical
experiments are performed to validate the accuracy and shock
capturing efficacy of these entropy stable DG methods.