Galerkin Method Research Articles

Two Finite Element Approaches for the Porous Medium Equation That Are Positivity Preserving and Energy Stable

In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

Preconditioned Nonsymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

Preconditioned Nonsymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

[formula omitted]-adaptive discontinuous Galerkin solution of transonic viscous flows with variable time step-size

The discontinuous Galerkin methods demonstrated to be well suited for scale resolving simulations of complex configurations, characterized by different fluid dynamics phenomena, such as transition, separation, shock/boundary layer interaction. Unfortunately, solvers based on these methods cannot yet reach the computational efficiency of well-established standard solvers, e.g., based on finite volume methods. To reduce the computational cost and the memory footprint, while not spoiling the accuracy, different approaches were investigated, which act on the spatial or the temporal discretization, and on the linear algebra. The above approaches have been extensively investigated, but rarely considering their mutual interactions. In this work, a p-adaptation algorithm is coupled with a time step-size adaptation algorithm to mitigate robustness issues arising after each adaptation cycle, probably related to the projection of the old solution on the new polynomial orders distribution. Moreover, this strategy is able to control the transient phase before each adaptation cycle, and can handle also the presence of initial steady solution, when the mesh is too coarse and the polynomial degree has not yet compensated for the lack of spatial accuracy. The effectiveness and accuracy of the proposed algorithm are demonstrated on transonic viscous flows with shock-wave/boundary-layer interaction.

Analyzing porous cold storage unit in presence of hybrid nano-powders considering Galerkin method

The article presents a comprehensive simulation study on cold storage porous containers, employing the Galerkin method to analyze their thermal behavior. A key focus of the investigation is the incorporation of both radiation and conduction modes within the mathematical model, acknowledging their significant roles in the freezing process. To enhance the freezing rate, hybrid nanoparticles are introduced into H2O, aiming to speed up the solidification. Notably, the study integrates adaptive grid techniques to achieve higher modeling accuracy, with verification processes confirming a high level of agreement between simulation results and experimental data. This research also underscores the potential for improved energy efficiency in cold storage systems, contributing to the sustainability of natural resources by reducing energy consumption and enhancing the overall efficiency of thermal storage systems. Through meticulous examination, the study explores the impacts of various factors on freezing, including the fraction of hybrid nano-powders (ϕ), porosity (γ), and the radiation factor (Rd). Remarkably, the introduction of porous media causes to a substantial increment in the freezing rate, resulting in an improvement of approximately 91.19 %. Moreover, the inclusion of the radiation term in the model significantly reduces the completion time by approximately 30.85 %, underscoring the importance of considering radiation effects in such systems. Additionally, dispersing hybrid nano-powders into the water, results in a decrement in the freezing time around 7.21 %. This approach highlights the significant potential for optimizing energy use in thermal storage applications, thereby promoting the conservation and sustainable management of natural resources.

Travelling waves in beam-like structures submerged in water

There are many examples in nature of travelling waves used for propulsion purposes, e.g., micro-organisms and sea creatures. Structural travelling waves can be used to induce momentum in a surrounding media creating a net propelling force. Recent research has tried to capture this interaction in engineering devices. Nonetheless, some challenges remain to fully exploit this phenomenon so that travelling-waves propelled devices can be optimally designed. One such challenge is that the interaction between the structure and the surrounding fluid heavily influences the amplitude of the waves and how they travel through the structure. This paper proposes a systematic qualitative and quantitative analysis of travelling waves in a slender cantilever beam submerged in water. The novelty of this work is demonstrated through two key aspects: The application of the Euler-Bernoulli beam equation combined with the Galerkin approximation, enabling a deeper understanding of how travelling waves form at resonant frequencies rather than non-resonant ones; and An analytical approach using a Galerkin approximation to characterise the nonlinear fluid-structure interaction, followed by linearisation for a comprehensive parametric study of the problem. In this investigation, the contributions of the first five vibration modes are considered in relation to the travelling waves observed near the resonant peaks. Experimental tests validate the analytical results and assess the accuracy of the proposed models. The results demonstrate that the model presented effectively characterises the travelling waves, making a suitable tool for the design of travelling-wave propelled devices.

Discontinuous Galerkin Methods for 3D–1D Systems

Discontinuous Galerkin Methods for 3D–1D Systems

The numerical solution of Fredholm integral equation of the first kind using the Sinc-Galerkin method

Sinc-Galerkin scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. In this article, a numerical solution of Fredholm integral equation of the first kind which arise in a large number of problems in applied mathematics will be discussed, for this result, we choose Sinc functions as basis functions and Galerkin method to estimate a solution for an unknown function in this equation. So, in this paper, some properties of the Sinc-Galerkin method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some linear algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to confirm applicability and justify rapid convergence of our method.

The impact of rotation on the onset of cellular convective movement in a casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations

In this effort, we examined the impact of rotation on the arrival of cellular convective motion in a Casson fluid saturated permeable layer with temperature dependent thermal conductivity and viscosity deviations. The problem is important to cellular foams prepared from plastics, ceramics, and metallic where radiation conductivity is revealed as a power function of temperature. The altered Darcy model is used to characterize the rheological performance of Casson fluid flow in permeable medium. The approximate analytical solution and numerical solution correct to one decimal place are presented utilizing the Galerkin method. The analysis reveals that the influence of thermal conductivity disparity parameter and the rotation is to delay the convective motion whereas; the viscosity disparity parameter and the Casson parameter have dual impact on the convective motion in the presence of rotation. The range of the convective cell drops with increasing the thermal conductivity disparity parameter, the viscosity disparity parameter, the Casson parameter and rotation parameter. In the absence of rotation, the range of the convective cell does not depend on the Casson parameter and the viscosity disparity parameter. Further, the existing results are compared with the existing literature under the particular case of this study.

Topology optimization for minimizing the mean compliance under thermo-mechanical loads using element-free Galerkin method

This paper presents a numerical model for addressing thermo-mechanical coupling problems in topology optimization, utilizing the element-free Galerkin (EFG) method. A multi-parameter density-based topology optimization framework is introduced to interpolate the thermal stress coefficient, encompassing thermal conductivity, thermal expansion coefficient, and elastic modulus. Two numerical examples are provided to investigate the model's characteristics and associated considerations, including the impact of EFG node distribution, quantity, calculation points layout, design variables, and filtering techniques on optimized results. Numerical findings indicates that the present model allows for adaptable adjustments in nodes number and distribution, calculation points layout, choice of design variables, and application of various filtering techniques while maintaining consistent background grids. Although these adjustments may affect convergence rates and final objective values, they can promise satisfactory optimization structures. Additionally, the study highlights the critical influence of filtering radius on optimization outcomes and the objective value, recommending a value of approximately 16∼20 calculation points.

A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method

AbstractThis article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

Phase-Informed Discontinuous Galerkin Method for Extremely High-Frequency Wave Modeling

Phase-Informed Discontinuous Galerkin Method for Extremely High-Frequency Wave Modeling

Martingale solutions and asymptotic behaviors for a stochastic cross-diffusion three-species food chain model with prey-taxis.

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis.

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.

Моделирование течения воздуха в упруго-деформируемой пористой среде, аппроксимирующей легкие человека: структура модели, ее основные уравнения и разрешающие соотношения

This paper is devoted to refining the mathematical model of the human respiratory system designed to predict the occurrence of respiratory pathologies caused by the negative effects of atmospheric air pollution. It is difficult to identify individual air channels in the human lungs, consisting of small respiratory tracts, called alveoli, through which oxygen from the air passes into blood. Therefore, the human lungs are modeled as a two-phase elastically deformable saturated porous medium, in which one phase is the deformable skeleton of the medium (lung tissue) described by the model of deformable solid body, and the other phase is gas filling the porous space. We construct the model of air flow in the human lungs, which consists of the sub-model of elastic deformation of the porous lung medium, experiencing large displacement gradients, and the sub-model of air filtration through the deformable porous medium. The model takes into account the relationships between sub-models of the respiratory system, i.e. we consider the coupled problems of deformation and filtration in differential form. Since at present the analytical solution of the stated essentially nonlinear problem is unfeasible, the problem is formulated in a generalized weak form to allow subsequent application of numerical methods using the Galerkin method. For the numerical solution of the nonlinear problem of deformation of the two-phase lung medium, it is suggested to use the finite element method, for the filtration problem - the finite volume method. The resolving relations needed to solve the formulated sub-problems are derived. To solve the nonlinear problem, we developed algorithms and a set of programs. The description of these computational tools, as well as analysis of the results obtained will be presented in our the next paper.

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

Vibration and acoustic characteristics of aluminium-silicon carbide metal matrix composite plate under thermal environment

This paper investigates the vibration and acoustic characteristics of an aluminium-silicon carbide (Al-SiC) simply supported composite plate under thermal loads. The vibrational response induced by the temperature rise is analysed using Kirchhoff’s plate theory and the weighted integral (Galerkin) method. Additionally, the plate is acoustically modelled in conjunction with the air medium to investigate its acoustic response. The acoustic far-field radiation equation determines the plate’s sound pressure level (SPL). The findings reveal that the MMC I (65% SiC/A356.2 MMC SiC particulates reinforced, with three different sizes) plate exhibits a lower overall sound pressure level in the lower frequency range than other MMCs. Moreover, replacing the aluminium plate with the MMC I plate reduces the overall sound pressure level by 5.54 dB due to the high stiffness associated with the mixture of silicon carbide reinforcement in the aluminium matrix.

Positive solution for a nonlinear elliptic equation on symmetric domains

We show the existence of a positive solution for the Schrodinger quasilinear equation with variable exponents above the critical regime. For that matter, we show an embedding into an Orlicz space of functions modeled over radially symmetric domains. Then we use a Galerkin method combined with a fixed-point argument to obtain a solution. For more information see https://ejde.math.txstate.edu/Volumes/2024/41/abstr.html