Volume 1, Issue 3
The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework

Rhodri L. T. Bevan, Raoul vanLoon & Perumal Nithiarasu

Adv. Appl. Math. Mech., 1 (2009), pp. 319-340.

Published online: 2009-01

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  • Abstract

This paper presents a comprehensive overview of the element-wise locally conservative Galerkin (LCG) method. The LCG method was developed to find a method that had the advantages of the discontinuous Galerkin methods, without the large computational and memory requirements. The initial application of the method is discussed, to the simple scalar transient convection-diffusion equation, along with its extension to the Navier-Stokes equations utilising the Characteristic Based Split (CBS) scheme. The element-by-element solution approach removes the standard finite element assembly necessity, with an face flux providing continuity between these elemental subdomains. This face flux provides explicit local conservation and can be determined via a simple small post-processing calculation. The LCG method obtains a unique solution from the elemental contributions through the use of simple averaging. It is shown within this paper that the LCG method provides equivalent solutions to the continuous (global) Galerkin method for both steady state and transient solutions. Several numerical examples are provided to demonstrate the abilities of the LCG method.

  • Keywords

Explicit local flux conservation, element-by-element solution, local conservation, LCG, convection-diffusion, CBS, artificial compressibility, incompressible flow.

  • AMS Subject Headings

35D99, 65-05, 76D99, 76Z05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-1-319, author = {}, title = {The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {3}, pages = {319--340}, abstract = {

This paper presents a comprehensive overview of the element-wise locally conservative Galerkin (LCG) method. The LCG method was developed to find a method that had the advantages of the discontinuous Galerkin methods, without the large computational and memory requirements. The initial application of the method is discussed, to the simple scalar transient convection-diffusion equation, along with its extension to the Navier-Stokes equations utilising the Characteristic Based Split (CBS) scheme. The element-by-element solution approach removes the standard finite element assembly necessity, with an face flux providing continuity between these elemental subdomains. This face flux provides explicit local conservation and can be determined via a simple small post-processing calculation. The LCG method obtains a unique solution from the elemental contributions through the use of simple averaging. It is shown within this paper that the LCG method provides equivalent solutions to the continuous (global) Galerkin method for both steady state and transient solutions. Several numerical examples are provided to demonstrate the abilities of the LCG method.

}, issn = {2075-1354}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aamm/8372.html} }
TY - JOUR T1 - The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 319 EP - 340 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aamm/8372.html KW - Explicit local flux conservation, element-by-element solution, local conservation, LCG, convection-diffusion, CBS, artificial compressibility, incompressible flow. AB -

This paper presents a comprehensive overview of the element-wise locally conservative Galerkin (LCG) method. The LCG method was developed to find a method that had the advantages of the discontinuous Galerkin methods, without the large computational and memory requirements. The initial application of the method is discussed, to the simple scalar transient convection-diffusion equation, along with its extension to the Navier-Stokes equations utilising the Characteristic Based Split (CBS) scheme. The element-by-element solution approach removes the standard finite element assembly necessity, with an face flux providing continuity between these elemental subdomains. This face flux provides explicit local conservation and can be determined via a simple small post-processing calculation. The LCG method obtains a unique solution from the elemental contributions through the use of simple averaging. It is shown within this paper that the LCG method provides equivalent solutions to the continuous (global) Galerkin method for both steady state and transient solutions. Several numerical examples are provided to demonstrate the abilities of the LCG method.

Rhodri L. T. Bevan, Raoul vanLoon & Perumal Nithiarasu. (1970). The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework. Advances in Applied Mathematics and Mechanics. 1 (3). 319-340. doi:
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