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Modelling convection dominated transport problems with improved Galerkin finite element formulations

  • Laitinen, Mikko
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The Galerkin finite element formulation is a widely used method for numerical solving of various engineering and scientific problems. In most cases the application of this method leads to symmetrical matrix equations which can be solved quite accurately. However, in certain cases the features of the pertinent problem considered result in nonsymmetric matrix equations and in these situations the Galerkin formulation has been observed to bring about considerable numerical difficulties. One example of this type of problems has been encountered in analysing the coupled mass transport equation when convective transport dominates over diffusive transport. In this case the classical Galerkin formulation is not capable of reproducing correctly sharp fronts in the concentration field and close to such boundaries strong oscillations in numerical values for concentration are observed. A transport equation of this kind has to be solved numerically when analysing the flow of groundwater in a domain where the salinity of the groundwater is varying. These type of situations can be relevant for certain potential locations for a repository of high-level nuclear waste deep in the crystalline bedrock. The modelling tools of VTT Energy have evolved during long-term development and application work. With the most recent version of the FEFLOW modelling system one is able to solve fully coupled equations for water flow, heat transfer and solute transport in a medium that can be described in each subdomain by the porous continuum approach. The report describes the latest modifications to the FEFLOW code system to account for the above described numerical instabilities. Based on a literature survey, a few most promising algorithms were chosen for a more detailed testing with practical examples concerning their effectiveness to damp the observed numerical instabilities with the conventional methods as well as their compatibility to the FEFLOW code system. From the results obtained it was concluded that among the methods studied the stream line upwind/Petrov Galerkin (SU/PG) formulation suits best to be applied with FEFLOW. This method effectively corrects the errors involved in the use of the classical Galerkin formulation to the situations described above. Furthermore, the method is also applicable in three-dimensional element meshes consisting of different types of elements.

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