Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations

Download
2014-09-04
In this work, we apply the adaptive discontinuous Galerkin (DGAFEM) method to the convection dominated non-linear, quasi-stationary diffusion convection reaction equations. We propose an efficient preconditioner using a matrix reordering scheme to solve the sparse linear systems iteratively arising from the discretized non-linear equations. Numerical examples demonstrate effectiveness of the DGAFEM to damp the spurious oscillations and resolve well the sharp layers occurring in convection dominated non-linear equations.
COMPUTERS & CHEMICAL ENGINEERING

Suggestions

Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations
Yücel, Hamdullah (2015-09-01)
We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau-Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a poste...
ADAPTIVE DISCONTINUOUS GALERKIN APPROXIMATION OF OPTIMAL CONTROL PROBLEMS GOVERNED BY TRANSIENT CONVECTION-DIFFUSION EQUATIONS
Yücel, Hamdullah; Benner, Peter (2018-01-01)
In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. ...
Discontinuous galerkin methods for time-dependent convection dominated optimal control problems
Akman, Tuğba; Karasözen, Bülent; Department of Scientific Computing (2011)
Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensi...
Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models
Yücel, Hamdullah; BENNER, Peter (2013-11-11)
In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component prob...
Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints
Yücel, Hamdullah (2013-08-30)
We study the numerical solution of control constrained optimal control problems governed by a system of convection diffusion equations with nonlinear reaction terms, arising from chemical processes. Control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches.
Citation Formats
M. UZUNCA, B. Karasözen, and M. Manguoğlu, “Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations,” COMPUTERS & CHEMICAL ENGINEERING, pp. 24–37, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/29859.