The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework (Advances in Applied Mathematics and Mechanics, 2018)
This paper presents a comprehensive overview of the element-wiselocally conservative Galerkin (LCG) method. The LCG method wasdeveloped to find a method that had the advantages of thediscontinuous Galerkin methods, without the large computational andmemory requirements. The initial application of the method isdiscussed, to the simple scalar transient convection-diffusionequation, along with its extension to the Navier-Stokes equationsutilising the Characteristic Based Split (CBS) scheme. Theelement-by-element solution approach removes the standard finiteelement assembly necessity, with an face flux providing continuitybetween these elemental subdomains. This face flux provides explicitlocal conservation and can be determined via a simple smallpost-processing calculation. The LCG method obtains a uniquesolution from the elemental contributions through the use of simpleaveraging. It is shown within this paper that the LCG methodprovides equivalent solutions to the continuous (global) Galerkinmethod for both steady state and transient solutions. Severalnumerical examples are provided to demonstrate the abilities of theLCG method.