We analyze the so-called the minimal dissipation local discontinuous Galerkin method (MD-LDG) for convection-diffusion or diffusion problems. The distinctive feature of this method is that the stabilization parameters associated with the numerical trace of the flux are identically equal to zero in the interior of the domain; this is why its dissipation is said to be minimal. We show that the orders of convergence of the approximations for the potential and the flux using polynomials of degree k are the same as those of all known discontinuous Galerkin methods, namely, (k + 1) and k, respectively. Our numerical results verify that these orders of convergence are sharp. The novelty of the analysis is that it bypasses a seemingly indispensable condition, namely, the positivity of the above mentioned stabilization parameters, by using a new, carefully defined projection tailored to the very definition of the numerical traces.
Bibliographical noteFunding Information:
The first author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute.
- Convection-diffusion equation
- Minimal dissipation local discontinuous Galerkin method