Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 11Citation - Scopus: 10A Stabilizing Subgrid for Convection-Diffusion Problem(World Scientific Publishing Co. Pte Ltd, 2006) Neslitürk, Ali İhsanA stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection-diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.Article Citation - WoS: 2Citation - Scopus: 3Bubble-Based Stabilized Finite Element Methods for Time-Dependent Convection–diffusion–reaction Problems(John Wiley and Sons Inc., 2016) Şendur, Ali; Neslitürk, Ali İhsanIn this paper, we propose a numerical algorithm for time-dependent convection–diffusion–reaction problems and compare its performance with the well-known numerical methods in the literature. Time discretization is performed by using fractional-step θ-scheme, while an economical form of the residual-free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases.Article Citation - WoS: 29Citation - Scopus: 35Finite Difference Approximations of Multidimensional Unsteady Convection-Diffusion Equations(Elsevier Ltd., 2015) Kaya, AdemIn this paper, the numerical approximation of unsteady convection-diffusion-reaction equations with finite difference method on a special grid is studied in the convection or reaction-dominated regime. We extend the method [19] which was designed for multidimensional steady convection-diffusion-reaction equations to unsteady problems. We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged). Discretization in time is performed by using Crank-Nicolson and Backward-Euler finite difference schemes, while for the space discretization we consider the method [19]. Numerical tests are presented to show good performance of the method.Article Citation - WoS: 10Citation - Scopus: 12Finite Difference Approximations of Multidimensional Convection-Diffusion Problems With Small Diffusion on a Special Grid(Elsevier Ltd., 2015) Kaya, Adem; Şendur, AliA numerical scheme for the convection-diffusion-reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind Petrov-Galerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.Article Citation - WoS: 15Citation - Scopus: 18A Finite Difference Scheme for Multidimensional Convection-Diffusion Equations(Elsevier, 2014) Kaya, AdemIn this paper a finite difference scheme is proposed for multidimensional convection-diffusion-reaction equations, particularly designed to treat the most interesting case of small diffusion. It is based closely on the work S¸endur and Neslitu¨rk (2011). Application of the method to multidimensional convection-diffusion-reaction equation is based on a simple splitting of the convection-diffusion-reaction equation and then joining their approximations obtained with S¸endur and Neslitu¨rk (2011). The method adapts very well to all regimes with continuous transitions from one regime to another. Numerical tests show good performance of the method and superiority with respect to well known stabilized finite element methods.Article Citation - WoS: 55Citation - Scopus: 56Two-Level Finite Element Method With a Stabilizing Subgrid for the Incompressible Mhd Equations(John Wiley and Sons Inc., 2010) Aydın, Selçuk Han; Neslitürk, Ali İhsan; Tezer Sezgin, MünevverWe consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate solutions obtained show the well-known characteristics of the MHD flow. Copyright © 2009 John Wiley & Sons, Ltd.Article Citation - WoS: 10Citation - Scopus: 10On the Choice of Stabilizing Sub-Grid for Convection-Diffusion Problem on Rectangular Grids(Elsevier Ltd., 2010) Neslitürk, Ali İhsanA stabilizing sub-grid which consists of a single additional node in each rectangular element is analyzed for solving the convection-diffusion problem, especially in the case of small diffusion. We provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method. We further study convergence properties of the method under consideration and prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the SUPG method. Some numerical experiments that confirm the theoretical findings are also presented. © 2010 Elsevier Ltd. All rights reserved.Article Citation - WoS: 37Citation - Scopus: 40Finite Element Method Solution of Electrically Driven Magnetohydrodynamic Flow(Elsevier Ltd., 2006) Neslitürk, Ali İhsan; Tezer, MünevverThe magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field.Article Citation - WoS: 66Citation - Scopus: 67The Finite Element Method for Mhd Flow at High Hartmann Numbers(Elsevier Ltd., 2005) Neslitürk, Ali İhsan; Tezer, MünevverA stabilized finite element method using the residual-free bubble functions (RFB) is proposed for solving the governing equations of steady magnetohydrodynamic duct flow. A distinguished feature of the RFB method is the resolving capability of high gradients near the layer regions without refining mesh. We show that the RFB method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the Hartmann number. Numerical results confirming theoretical findings are presented for several configurations of interest. The approximate solution obtained by the RFB method is also compared with the analytical solution of Shercliff's problem.Article Citation - WoS: 15Citation - Scopus: 16The Nearly-Optimal Petrov-Galerkin Method for Convection-Diffusion Problems(Elsevier Ltd., 2003) Neslitürk, Ali İhsan; Harari, IsaacThe nearly-optimal Petrov-Galerkin (NOPG) method is employed to improve finite element computation of convection-dominated transport phenomena. The design of the NOPG method for convection-diffusion is based on consideration of the advective limit. Nonetheless, the resulting method is applicable to the entire admissible range of problem parameters. An investigation of the stability properties of this method leads to a coercivity inequality. The convergence features of the NOPG method for convection-diffusion are studied in an error analysis that is based on the stability estimates. The proposed method compares favorably to the performance of an established technique on several numerical tests.