Homogenization of the transport equation describing convection-diffusion processes in a material with fine periodic structure

Šilhánek, David, and Beneš, Michal. "Homogenization of the transport equation describing convection-diffusion processes in a material with fine periodic structure." Programs and Algorithms of Numerical Mathematics. Prague: Institute of Mathematics CAS, 2023. 239-248. <http://eudml.org/doc/299001>.

@inProceedings{Šilhánek2023,
abstract = {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.},
author = {Šilhánek, David, Beneš, Michal},
booktitle = {Programs and Algorithms of Numerical Mathematics},
keywords = {variational principles; homogenization; $\Gamma $-convergence; convection-diffusion equation; optimal artificial diffusion},
location = {Prague},
pages = {239-248},
publisher = {Institute of Mathematics CAS},
title = {Homogenization of the transport equation describing convection-diffusion processes in a material with fine periodic structure},
url = {http://eudml.org/doc/299001},
year = {2023},
}

TY - CLSWK
AU - Šilhánek, David
AU - Beneš, Michal
TI - Homogenization of the transport equation describing convection-diffusion processes in a material with fine periodic structure
T2 - Programs and Algorithms of Numerical Mathematics
PY - 2023
CY - Prague
PB - Institute of Mathematics CAS
SP - 239
EP - 248
AB - 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.
KW - variational principles; homogenization; $\Gamma $-convergence; convection-diffusion equation; optimal artificial diffusion
UR - http://eudml.org/doc/299001
ER -