It 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).

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 PetrovGalerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.

A 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 L 1 -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.

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 convectiondominated 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.

We propose a fully discrete -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 -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 . Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.

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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. Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not Anahtar kelimeler Av-avcı denklem sistemleri; Konveksiyon-difüzyonreaksiyon; Kararlı Sonlu Eleman Yöntemi; Çok-ölçekli Yöntemler. Öz Bu çalışmada, konveksiyon-difüzyon-reaksiyon problemleri ile modellenebilen av-avcı denklem sistemlerinin simülasyonunda kullanılan sayısal çözüm tekniklerini iyileştirecek ve daha etkin sonuçlar üretecek sayısal bir yöntem önerilmiştir. Uzay ayrıklaştırması için, sonlu elemanlar metodunu uygularken seçilen polinom baz fonksiyonlarına ilaveten fonksiyon uzayının özel tip fonksiyonlarla (residual-free bubbles) zenginleştirilmesine dayanan Pseudo Residual-free Bubble (PRFB) yöntemi kullanılmıştır. Söz konusu yöntem, çeşitli test örneklerine uygulanmış olup elde edilen sayısal çözümlerin, literatürde mevcut olan sonuçlar ile iyi bir uyum içinde olduğu gözlemlenmiştir. Sayısal sonuçlar, önerilen yöntemin verimli ve uygulanabilir olduğunu göstermektedir.

Aim: Many people around the world are still affected by coronavirus disease. The high spread rate of the COVID-19 and the number of deaths make it more important than ever to understand the behavior of the disease. In this work, the effects of health policies on the COVID-19 process is examined by taking into account various data.Material and Methods: Official articles, circulars issued by decision makers, Worldometer and Our World in Data websites are used to study the effects of the health policies on the COVID-19 process from 10.03.2020 to 30.06.2021. The scenario in which the appropriate precautions were not taken by decision makers is also considered by simulating data with the method of least squares.Results: It has been determined that the measures taken have positive effects with a delay of 12-15 days. We observe that the policies adopted for the COVID-19 process are effective in reducing the number of cases and deaths. We note that there were increases in the number of cases after the removal of some measures. Overall, Turkey's situation in terms of total case and mortality rates in COVID-19 process is better than the average of some European and neighboring countries. Conclusion:The results shows that Turkey has quite disciplined and successful health policy to overcome coronavirus disease.