Saddle point least squares and Petrov-Galerkin methods applied to reaction-diffusion and convection-diffusion equations
Date
2023
Authors
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Publisher
University of Delaware
Abstract
We present new results on both saddle point least-squares and Petrov-Galerkin methods applied to reaction-diffusion and convection-diffusion equations. In the reaction-diffusion equation, we provide a preconditioner which proves to be efficient and easily implementable in practice. For the convection-diffusion equation, we present an optimal trial norm for both a saddle point least-squares reformulation, as well as an upwinding Petrov-Galerkin method. For the Petrov-Galerkin method in one dimension, we make a connection to a standard streamline upwind Petrov-Galerkin method. We introduce optimal trial norms that enable robust stability analysis of the discretization approach. Furthermore, our discretization methods provide efficient and trustworthy means of approximation. For the two dimensional convection-diffusion equation, we present three methods of discretization. These methods are natural extensions of ideas of the one dimensional problems leading to stabilization and accurate approximation. Numerical results are included to support the theoretical aspects, as well as to give motivation for directions of further research and development.
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Keywords
Saddle point least-squares, Petrov-Galerkin method, Convection-diffusion equations, Optimal trial norm