Abstract
In this book adaptive algorithms are developed for efficient discretization of advection dominated stationary and non-stationary semi-linear ADR equations. In order to handle the unphysical oscillations due to the advection, we have applied a symmetric interior penalty Galerkin (SIPG) method as an alternative to the well-known stabilized continuous FEM methods such as the streamlined upwind Petrov-Galerkin (SUPG) method. We have given a detailed construction of SIPG formulation on the general Poisson equation, and we have discussed the effect of the penalty parameter in Chapter 2. In Chapter 3, we gave existence and uniqueness results for stationary semi-linear ADR equations. We have shown that the space-time adaptive algorithm is robust and can resolve not only the layers produced by advection but also the sharp fronts due to the non-linear reaction as an alternate to the shock/discontinuity capturing techniques in the literature.We have also shown that adaptive dG approximations for stationary problems are more accurate than the Galerkin least squares FEMs and shock/discontinuity capturing techniques. Moreover, we have introduced an efficient iterative method, matrix reordering technique, as a preconditioner to solve the linear systems arising from the Newton’s method applied to the discrete system of stationary semi-linear ADR equations.
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© 2016 Springer International Publishing Switzerland
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Uzunca, M. (2016). Conclusions and Outline. In: Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30130-3_5
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DOI: https://doi.org/10.1007/978-3-319-30130-3_5
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-30129-7
Online ISBN: 978-3-319-30130-3
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