Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article On Smoothers for Multigrid of the Second Kind(John Wiley and Sons Inc., 2019) Aksoylu, Burak; Kaya, AdemWe study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nystrom method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi-like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted-Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.Article Citation - WoS: 5Citation - Scopus: 10Conditioning and Error Analysis of Nonlocal Operators With Local Boundary Conditions(Elsevier Ltd., 2018) Aksoylu, Burak; Kaya, AdemWe study the conditioning and error analysis of novel nonlocal operators in 1D with local boundary conditions. These operators are used, for instance, in peridynamics (PD) and nonlocal diffusion. The original PD operator uses nonlocal boundary conditions (BC). The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local periodic, antiperiodic, Neumann, or Dirichlet BC. We prove sharp bounds for their condition numbers in the parameter δ only, the size of nonlocality. We accomplish sharpness both rigorously and numerically. We also present an error analysis in which we use the Nyström method with the trapezoidal rule for discretization. Using the sharp bounds, we prove that the error bound scales like O(h2δ−2) and verify the bound numerically. The conditioning analysis of the original PD operator was studied by Aksoylu and Unlu (2014). For that operator, we had to resort to a discretized form because we did not have access to the eigenvalues of the analytic operator. Due to analytical construction, we now have direct access to the explicit expression of the eigenvalues of the novel operators in terms of δ. This gives us a big advantage in finding sharp bounds for the condition number without going to a discretized form and makes our analysis easily accessible. We prove that the novel operators have ill-conditioning indicated by δ−2 sharp bounds. For the original PD operator, we had proved the similar δ−2 ill-conditioning when the mesh size approaches 0. From the conditioning perspective, we conclude that the modification made to the original PD operator to obtain the novel operators that accommodate local BC is minor. Furthermore, the sharp δ−2 bounds shed light on the role of δ in nonlocal problems.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: 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: 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.