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variational principles; homogenization; $\Gamma$-convergence; convection-diffusion equation; optimal artificial diffusion
In the present contribution we discuss mathematical homogenization and numerical solution of the elliptic problem describing convection-diffusion processes in a material with fine periodic structure. Transport processes such as heat conduction or transport of contaminants through porous media are typically associated with convection-diffusion equations. It is well known that the application of the classical Galerkin finite element method is inappropriate in this case since the discrete solution is usually globally affected by spurious oscillations. Therefore, great care should be taken in developing stable numerical formulations. We describe a variational principle for the convection-diffusion problem with rapidly oscillating coefficients and formulate the corresponding homogenization results. Further, based on the variational principle, we derive a stable numerical scheme for the corresponding homogenized problem. A numerical example will be solved to illustrate the overall performance of the proposed method.
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