Balanced Norms Research Articles

A Finite Element Analysis in Balanced Norms for a Coupled System of Singularly Perturbed Reaction-Diffusion Equations

Abstract In this paper we study approximations of a singularly perturbed system of two coupled reaction-diffusion equations, in one dimension, by using piecewise linear finite elements on graded meshes. When the parameters are of different magnitudes, the solution exhibits in general two distinct but overlapping boundary layers. We prove that, when the mesh grading parameter is appropriately chosen, optimal error estimates in a balanced norm for piecewise linear elements can be obtained. Supporting numerical results are also presented.

A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

Do balanced budget practices of U.S. states make sense? Alternatives from the Eurozone

Purpose The purpose of this article is to challenge the balanced budget practices of U.S. state governments and offer alternatives that may lead to better fiscal, economic and policy outcomes. We contend that the norm of balance may be leading U.S. states to make fiscal decisions that result in less-than-ideal outcomes, especially during economic downturns.Design/methodology/approach This is a normative article. We examine the scholarly evidence regarding balanced budget practices to assess the appropriateness of balanced budget norms. We also examine the fiscal rules followed by Eurozone countries to draw potential lessons for U.S. states.Findings We conclude that state governments should move away from strict norms of budget balance and seek more flexible approaches. We suggest that instead of following strict rules and norms of balance, U.S. states should consider implementing escape clauses, debt and deficit ceilings, and fiscal councils. We also suggest that the Federal Reserve be open to lending directly to states during fiscal crises to ensure that states have access to affordable credit.Originality/value The balanced budget norm has become ingrained in U.S. state budgeting practices, so much so that public officials and scholars alike rarely question it. The novel contribution of our article is to question this practice in a systematic way and propose alternative approaches.

The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k≥1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ−uniform error bound of O(N−(k+1)+ɛ(N−1lnN)k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ−uniform error bound of O(N−(k+1/2)+(N−1lnN)k) in the balanced norm is established by the weighted local L2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O((N−1lnN)k+1) in the L2 norm and the discrete L∞ norm uniformly with respect to the singular perturbation parameter ɛ.

The Level of Sex Hormones and Corticosterone in Female Rats during Modeling of Visceral Obesity, Subsequent Physical Activity, and Normalization of the Diet.

The effects of diet-induced visceral obesity and non-drug options of its correction on the level of sex hormones and corticosterone were evaluated in 84 female Wistar rats. During stage I, the rats received either a standard diet (STD) or a high-calorie diet (HCD) for 8 weeks. During stage II, the animals were divided into subgroups depending on obesity correction: without correction (STD control and HCD), transition from HCD to STD (HCD/STD) and/or physical activity (treadmill exercise) for the next 8 weeks (HCD/STD+exercise, STD+exercise, and HCD+exercise). Diet-induced visceral obesity resulted in hyperandrogenization and increased blood corticosterone levels in females. Transition from HCD to STD regardless of physical activity led to normalization of testosterone level and, accordingly, to return to the functional norm of estrogen-androgen balance. The positive effect of moderate physical activity on hormonal status is realized only against the background of a balanced diet or during the transition from HCD to STD.

A balanced finite-element method for an axisymmetrically loaded thin shell

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.

Error analysis of a weak Galerkin finite element method for singularly perturbed differential-difference equations

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings.

Temporal Framing in Balanced News Coverage of Artificial Intelligence and Public Attitudes

ABSTRACT This study investigates how balanced news coverage, presenting both the positive and negative aspects of artificial intelligence (AI), may lead to attitudinal changes among the public when it is paired with different temporal frames. Different construal levels associated with temporal distance and evaluations of pros and cons provide a theoretical grounding to understand these dynamics. The results of an online experiment showed that exposure to the near-future temporal framing of AI (vs. distant-future temporal framing) induced greater perceived severity of AI risks. This, in turn, decreased public support for AI development and use. The indirect effect was conditional upon message order such that the effect was only significant when the message was presented in the risk/benefit order. The findings suggest that non-refutational two-sided frames reflecting journalistic norms of balance may lead to public opinion change because the accompanying temporal frames unwittingly impact which side people put more weight on. Thus, the selection of sources and the use of time reference need to be carefully vetted by journalists when reporting controversial science and technology.

Error Estimate in the Balanced Norm for the Singularly Perturbed Reaction–Diffusion Problems

Error Estimate in the Balanced Norm for the Singularly Perturbed Reaction–Diffusion Problems

Supercloseness and post-processing of finite element method in a balanced norm for singularly perturbed reaction-diffusion equation

Supercloseness and post-processing of finite element method in a balanced norm for singularly perturbed reaction-diffusion equation

Influence of calcium nitrate on potato productivity and soil condition in the forest-steppe of the Chelyabinsk region

The article discusses the results of field experiments to study the effect of calcium on the formation of the planned yield and quality of potato tubers in the conditions of the Chelyabinsk region. The objectives of the study included assessing the effect of mineral fertilizers on the agrochemical properties of the soil. It has been proven that the use of balanced norms of mineral fertilizers based on a planned yield of 40 t/ha (both with calcium and without it) has a significant impact on the mineral nutrition regime of plants. The introduction of calcium increases the adaptive ability of plants to drought conditions, while an increase in nitrogen nutrition of potatoes is observed in the first half of the growing season, and phosphorus and potassium nutrition throughout the growing season. As a result, in the variant (N130P150K130Ca158) with the introduction of calcium, the highest yield (+41.3% of the control) and the marketability of tubers (+5.1%), as well as high product quality, are observed, while in the variant without calcium (N130P150K130) a decrease in the content of dry matter in tubers by 1.7%, and starch - by 0.99% compared with the control (without fertilizers).

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

For the error analysis of singularly perturbed reaction-diffusion problems, the balanced norm, which is stronger than the usual energy norm, is introduced to correctly reflect the behavior of the layer. In this paper, we study the convergence in a balanced norm for nonsymmetric interior penalty Galerkin (NIPG) method for the first time. For this purpose, a new interpolation is designed, which consists of a Gauß Lobatto interpolation in the layer, and a locally weighted L2 projection outside the layer. On that basis, by properly defining the penalty parameters at different nodes on a Shishkin mesh, we obtain the supercloseness of almost k+12 order, and prove the convergence of optimal order in a balanced norm. Here k is the degree of polynomials. Numerical experiments verify the main conclusion.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

FEATURES OF DIAGNOSIS AND TREATMENT OF CROHN’S DISEASE (CLINICAL CASE)

The exact reason for the occurrence of Crohn’s disease remains unknown. It is believed that pathology can be carried out by infectious and viral pathogens, deviations from the norm of natural balance of the microflora of the gastrointestinal tract (hereinafter – the gastrointestinal tract). Many researchers indicate an abnormal autoimmune response of the body on food stimuli – in favor of such an assertion indicates the systematic impression of the internal organs. However, for the emergence and development of this disease, there are significant impacts with harmful habits, such as: smoking, alcoholic beverages and medicines, unbalanced nutrition and the presence of chronic inflammatory gastroids. The inflammatory process begins in the mucous membrane, gradually affects all layers of the wall of the gastrointestinal wall, leads to its destruction and fibrosis and the formation of fistish and stenoses. Therefore, early diagnosis and timely treatment prevents the development of complications, including such severe such as bleeding, perforation of the intestinal wall, intra-abdominal abscesses, internal and external fists, adhesions of the intestine with the development of obstruction. Purpose: on the basis of a specific case of HC to investigate and analyze the features of diagnosis and treatment of these patients using an analytical method and a systematic approach. Materials and methods: The material was a clinical case of HC, methods of instrumental diagnosis and objective analysis. The results: The concrete case of HC is investigated, features of inspection and diagnostics are defined. Conclusions: A feature of the diagnosis and treatment of patients with HC is a constant vigilance about the occurrence of complications, the key to the prevention of which is a multidisciplinary approach in the tactics of diagnosis and treatment of these patients. Key words: Crohn’s disease, features of examination and diagnosis.

Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms

A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results.

Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on Bakhvalov-type meshes for nonlinear singularly perturbed problems

Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on Bakhvalov-type meshes for nonlinear singularly perturbed problems

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

Analysis of finite element method in balanced norms for two-parameter singularly perturbed problems

We consider the two-parameter singularly perturbed problems, and there are two exponential boundary layers in its solution. In general, the energy norm is used in error estimations of finite element method for singularly perturbed problems. However, it is difficult to capture each layer simultaneously in this norm. In this paper, a special balanced norm is defined, which can capture each exponential layer well. Based on the balanced norm, we obtain the uniform convergence of any order finite element method with respect to both parameters on a Shishkin mesh. Moreover, the optimal order is proved and some discussions in this balanced norm are presented. Finally, we give some numerical examples to verify our theoretical findings.

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.