Yazar "Şendur, Ali" seçeneğine göre listele

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    A comparative study on stabilized finite element methods for the convection-diffusion-reaction problems
    (Hindawi Limited, 2018) Şendur, Ali
    The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convection-dominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations. © 2018 Ali Sendur.
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    A note on epidemiologic models: SIR modeling of the COVID-19 with variable coefficients
    (Karaganda Üniversitesi-Matematik Bülteni, 2022) Çakır, Zafer; Şendur, Ali
    The coronavirus disease 2019 (COVID-19) has been responsible for over three million reported cases worldwide. The construction of an appropriate mathematical (epidemiological) model for this disease is a challenging task. In this paper, we first consider susceptible — infectious — recovered (SIR) model with constant parameters and obtain an approximate solution for the SIR model with varying coefficient as it is one of the simplest models and many models are derived from this framework. The numerical experiments confirm that the proposed formulation demonstrates similar characteristic behaviour with the well-known approximations.
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    A note on stabilized finite element methods for predator-prey systems
    (2019) Şendur, Ali
    A numerical method that will improve and produce effective results for solving mathematical model for the system of predator-prey interactions which is defined by convection-diffusion-reaction problem is studied herein. We consider the Pseudo Residual-free Bubble (PRFB) method which is based on augmenting the finite element space by appropriate functions for the space discretization. The method is applied on different test problems and the numerical solutions are in good agreement with the result available in literature. The numerical results depict that the algorithm is efficient and feasible.
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    A stabilizing augmented grid for rectangular discretizations of the convection-diffusion-reaction problems
    (Springer-Verlag Italia Srl, 2018) Şendur, Ali
    We propose a numerical method for approximate solution of the convection-diffusion-reaction problems in the case of small diffusion. The method is based on the standard Galerkin finite element method on an extended space defined on the original grid plus a subgrid, where the original grid consists of rectangular elements. On each rectangular elements, we construct a subgrid with few points whose locations are critical for the stabilization of the problem, therefore they are chosen specially depending on some specific conditions that depend on the problem data. The resulting subgrid is combined with the initial coarse mesh, eventually, to solve the problem in the framework of Galerkin method on the augmented grid. The results of the numerical experiments confirm that the proposed method shows similar stability features with the well-known stabilized methods for the critical range of problem parameters.
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    Bubble-based stabilized finite element methods for time-dependent convection-diffusion-reaction problems
    (Wiley, 2016) Şendur, Ali; Neslitürk, Ali İhsan
    In 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 theta-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. Copyright (C) 2016 John Wiley & Sons, Ltd.
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    Error estimates for a fully discrete epsilon-uniform finite element method on quasi uniform meshes
    (2021) Şendur, Ali; Natesan, Srinivasan; Singh, Gautam
    In this article, we analyze a fully discrete epsilon-uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, we choose L-splines as basis functions. We will concentrate on the convergence analysis of the finite element method which employ the discrete L-spline basis functions instead of their continuous counterparts. The L-splines are approximated on the piecewise-uniform Shishkin mesh inside each element. These approximations are used as basis functions in the frame of Galerkin FEM on a coarse piecewise-uniform mesh to discretize the domain. Further, we determine the amount of error introduced by the discrete L-spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods using the exact L-splines.
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    Error estimates for a fully discrete varepsilon?uniform finite element methodon quasi uniform meshes
    (2021) Şendur, Ali; Natesan, Srinivasan; Singh, Gautam
    In this article, we analyze a fully discrete varepsilon?uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, we choose L?splines as basis functions.We will concentrate on the convergence analysis of thefinite element methodwhich employ the discreteL?spline basis functions instead of their continuous counterparts.The L?splines are approximated on the piecewise-uniform Shishkin mesh inside each element. These approximations are used as basis functions in the frame of Galerkin FEM on a coarse piecewise-uniform meshto discretize the domain. Further, we determinethe amount of error introduced by the discrete L?spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods usingthe exact L?splines.
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    Finite difference approximations of multidimensional convection-diffusion-reaction problems with small diffusion on a special grid
    (Academic Press Inc Elsevier Science, 2015) Kaya, Adem; Şendur, Ali
    A 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. (C) 2015 Elsevier Inc. All rights reserved.