Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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: 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: 10Citation - Scopus: 12Applications of the Pseudo Residual-Free Bubbles To the Stabilization of the Convection-Diffusion Problems in 2d(Elsevier Ltd., 2014) Şendur, Ali; Neslitürk, Ali İhsan; Kaya, AdemA stabilized finite element method is studied herein for two-dimensional convection-diffusion-reaction problems. The method is based on the residual-free bubbles (RFB) method. However we replace the RFB functions by their cheap, yet efficient approximations computed on a specially chosen subgrid, which retain the same qualitative behavior. Since the correct spot of subgrid points plays a crucial role in the approximation, it is important to determine their optimal locations, which we do it through a minimization process with respect to the L1-norm. The resulting numerical method has similar stability features with the well-known stabilized methods in the literature for the whole range of problem parameters and this fact is also confirmed by numerical experiments.Article Citation - WoS: 9Citation - Scopus: 10Applications of the Pseudo Residual-Free Bubbles To the Stabilization of Convection-Diffusion Problems(Springer Verlag, 2012) Şendur, Ali; Neslitürk, Ali İhsanIt is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003). © 2011 Springer-Verlag.Article Citation - WoS: 1Citation - Scopus: 1A Fully Discrete ?-Uniform Method for Singular Perturbation Problems on Equidistant Meshes(Taylor and Francis Ltd., 2012) Filiz, Ali; Neslitürk, Ali; Şendur, AliWe propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally. © 2012 Copyright Taylor and Francis Group, LLC