A stabilised finite element method for the convection-diffusion-reaction equation in mixed form

This paper is devoted to the approximation of the convection-diffusion-reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems

On finite element methods for 3D time–dependent convection–diffusion–reaction equations with small diffusion

The paper studies finite element methods for the simulation of time–dependent convection–diffusion–reaction equations with small diffusion: the SUPG method, a SOLD method and two types of FEM–FCT methods. The methods are assessed, in particular with respect to the size of the spurious oscillations in the computed solutions, at a 3D example with nonhomogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions

Mixed formulations for the convection-diffusion equation

This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG.This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG

Stabilized finite element methods for time dependent convection-diffusion equations

Thesis (Doctoral)--Izmir Institute of Technology, Mathematics, Izmir, 2012Includes bibliographical references (leaves: 92-96)Text in English; Abstract: Turkish and Englishx, 96 leavesIn this thesis, enriched finite element methods are presented for both steady and unsteady convection diffusion equations. For the unsteady case, we follow the method of lines approach that consists of first discretizing in space and then use some time integrator to solve the resulting system of ordinary differential equation. Discretization in time is performed by the generalized Euler finite difference scheme, while for the space discretization the streamline upwind Petrov-Galerkin (SUPG), the Residual free bubble (RFB), the more recent multiscale (MS) and specific combination of RFB with MS (MIX) methods are considered. To apply the RFB and the MS methods, the steady local problem, which is as complicated as the original steady equation, should be solved in each element. That requirement makes these methods quite expensive especially for two dimensional problems. In order to overcome that drawback the pseudo approximation techniques, which employ only a few nodes in each element, are used. Next, for the unsteady problem a proper adaptation recipe, including these approximations combined with the generalized Euler time discretization, is described. For piecewise linear finite element discretization on triangular grid, the SUPG method is used. Then we derive an efficient stability parameter by examining the relation of the RFB and the SUPG methods. Stability and convergence analysis of the SUPG method applied to the unsteady problem is obtained by extending the Burman’s analysis techniques for the pure convection problem. We also suggest a novel operator splitting strategy for the transport equations with nonlinear reaction term. As a result two subproblems are obtained. One of which we may apply using the SUPG stabilization while the other equation can be solved analytically. Lastly, numerical experiments are presented to illustrate the good performance of the method

A posteriori optimization of parameters in stabilized methods for convection-diffusion problems --- Part I

Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing the stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator and error indicator. Benefits of the basic approach are shown and further improvements are discussed