Weak Elements Research Articles

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

A defect correction weak Galerkin finite element method for the Kelvin–Voigt viscoelastic fluid flow model

In this paper, we present a defect correction weak Galerkin finite element method for solving the Kelvin–Voigt viscoelastic fluid flow model, which can guarantee that the discrete velocity is globally divergence-free. The second-order Crank–Nicolson difference scheme is considered for temporal discretization. In particular, combined with the defect correction method, the numerical oscillation effect is reduced at high Reynolds number. Our algorithm not only gives the stability and convergence of the numerical solutions of velocity and pressure but also is uniformly valid at the retardation time κ→0. Several numerical experiments are performed to verify the method has good behaviors.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

Electromechanical coupling analysis of geometrically exact functionally graded piezoelectric shells based on weak form quadrature element method

Electromechanical coupling analysis of geometrically exact functionally graded piezoelectric shells based on weak form quadrature element method

Optimal [formula omitted] error estimates of stabilizer-free weak Galerkin finite element method for the drift-diffusion problem

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise Pk(k≥1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [Pj(K)]2(j≥j0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L2 projection operators, optimal error estimates of O(Δt+hk+1) in the L2 norm and O(Δt+hk) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis.

Weak Galerkin finite element methods for semilinear Klein–Gordon equation on polygonal meshes

Weak Galerkin finite element methods for semilinear Klein–Gordon equation on polygonal meshes

Weak Galerkin finite element methods for optimal control problems governed by second order elliptic equations

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

Analysis and computation of a weak Galerkin scheme for solving the 2D/3D stationary Stokes interface problems with high-order elements

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d , d = 2, 3. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ⩾ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

AbstractIn this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.

Very weak finite element methods: discretisation and applications

PurposeThis study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.Design/methodology/approachWe use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.FindingsVarious conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.Originality/valueThis is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

Yeast heterochromatin stably silences only weak regulatory elements by altering burst duration

Transcriptional silencing in Saccharomyces cerevisiae involves the generation of a chromatin state that stably represses transcription. Using multiple reporter assays, a diverse set of upstream activating sequence enhancers and core promoters were investigated for their susceptibility to silencing. We show that heterochromatin stably silences only weak and stress-induced regulatory elements but is unable to stably repress housekeeping gene regulatory elements, and the partial repression of these elements did not result in bistable expression states. Permutation analysis of enhancers and promoters indicates that both elements are targets of repression. Chromatin remodelers help specific regulatory elements to resist repression, most probably by altering nucleosome mobility and changing transcription burst duration. The strong enhancers/promoters can be repressed if silencer-bound Sir1 is increased. Together, our data suggest that the heterochromatic locus has been optimized to stably silence the weak mating-type gene regulatory elements but not strong housekeeping gene regulatory sequences.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Numerical modeling of the fracturing mechanisms of unconsolidated sand reservoirs under water injection

Produced Water Re-Injection is a standard practice in oil and gas operations. When performed at sufficiently high pressures, it can trigger the fracturing of the reservoir, which should be controlled to stimulate the formation without compromising its safety. While the mechanisms of the hydraulic fracturing of brittle rocks are well-understood, this understanding is more challenging in the case of unconsolidated sand reservoirs. Recent experimental studies indicate that pseudofractures in such formations primarily result from shear banding and flow channelization. There is a need for further interpretation of the experimental findings through numerical models that account for the proper physical phenomena. For that, a coupled finite element model of water percolation, particle transport and strain localization is developed to replicate the experiments of radial injection of water in mixtures of sand and fines by Nguyen et al., 2022: An experimental setup with radial injection cell for investigation of fracturing in unconsolidated sand reservoirs under fluid injection. Journal of Petroleum Science and Engineering 213:11036. To simulate shear banding, the model accounts for strain-softening behavior and introduces material imperfections in a few weak elements. Particle mobilization starts once a critical fluid velocity is reached and permeability is assumed to be a function of particle concentration. The numerical models can replicate the geometry of the observed radial pseudofractures, as well as the measured fracturing pressures. They also qualitatively capture the effects of the initial stress state on the fracturing pressure, the fracture length and on the permeability increases during the fracturing regime as observed in the laboratory tests. Notably, the numerical results gave hints on the role of the coupling between strain localization and particle transport on the formation of the pseudofractures. These findings are used to propose an updated conceptual model for the hydraulic fracturing of sand packs.

Semantic differentiation in the linguistic examination of trademarks

INTRODUCTION. The controversial issues of trademark examination are considered. The analysis of the ways of comparing single-word and compound commercial designations is given. A universal method for identifying a strong element in a composite designation has not yet been developed. The purpose of the study is to identify the semantic potential of non-identical concepts: a strong element and the main word in the phrase of a trademark or commercial designation.MATERIALS AND METHODS. Using a specific example from the experience of conducting linguistic examinations (comparison of the WHITE AURORA and AVRORA designations), the methodology for identifying a strong element of a composite designation is illustrated and the origins of typical errors in determining a strong element are analyzed. When developing the mentioned methodology, it is proposed to be guided by the following considerations: the concept of a strong element of designation and the main word of the corresponding phrase are concepts that are not common; the method of identifying a strong element in a phrase differs from the method of identifying the main word in a phrase; the phenomenon of a strong element correlates with the linguistic idea of a strong and weak lexical position in the text; a strong element can be located both in the first and second place in a phrase, including a trademark or other commercial designation.RESULTS AND DISCUSSION. It is shown that due to the semantic potential of a strong element of a composite designation, it becomes possible to transfer to the consumer the necessary information about the object of nomination. That is why the definition of an element bearing the main disti nguishing features must be carried out in the context of a situation or a communicative task. Based on the material of a specific example of linguistic expertise, it is shown that the element white of the designation WHITE AURORA in the context of the provision of dental services and dental care services can be interpreted as a strong element. This becomes a serious factor preventing the appearance of confusingly similar designations for WHITE AURORA on the one hand, and AVRORA on the other.CONCLUSION. Semantic differentiation is the main criterion for distinguishing strong and weak elements in the linguistic examination of trademarks. The very concept of a “strong element” of a designation can be considered a relative concept; the definition of an element bearing the main semantic features must be carried out in the context of a situation or a communicative task.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

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.

A discrete-ordinate weak Galerkin method for radiative transfer equation

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A priori error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

Generalized weak Galerkin finite element methods for second order elliptic problems

This article proposes and analyzes the generalized weak Galerkin (gWG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the gWG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual L2 norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed gWG method.