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**An element-wise, locally conservative Galerkin (LCG) method for solving diffusion and convection-diffusion problems.**
*(English)*
Zbl 1262.76059

Summary: An element-wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection-diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element-wise flux conservation. In the LCG method, elements are treated as sub-domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus,the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element.

Several problems of diffusion and convection-diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov-Galerkin weighting and high-order characteristic-based temporal schemes have been implemented into the LCG formulation.

Several problems of diffusion and convection-diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov-Galerkin weighting and high-order characteristic-based temporal schemes have been implemented into the LCG formulation.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76R99 | Diffusion and convection |

### Keywords:

explicit local flux conservation; element-by-element solution; heat conduction; convection-diffusion; characteristic-Galerkin; SUPG; linear and quadratic finite elements
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\textit{C. G. Thomas} and \textit{P. Nithiarasu}, Int. J. Numer. Methods Eng. 73, No. 5, 642--664 (2008; Zbl 1262.76059)

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