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In this talk, we consider new finite element methods for solving two different problems. One is coupled flow and transport systems in porous media and the other one is linear elasticity (mechanics) equation. The primary purpose of the study is to develop computationally efficient and robust numerical methods that could be free of both oscillations due to lack of local conservation and locking effects. The locally conservative enriched Galerkin (LF-EG), which will be utilized for solving flow problems is constructed by adding a constant function to each element based on the classical continuous Galerkin methods. The locking-free enriched Galerkin (LC-EG) adds a vector to the displacement space. These EG methods employs the well-known discontinuous Galerkin (DG) techniques, but the approximation spaces have fewer degrees of freedom than those for the typical DG methods, thus offering an efficient alternative to DG methods. We present a priori error estimates of optimal order. We also demonstrate through some numerical examples that the new method is free of oscillations and locking.
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