Search

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality. Bibliography: 57 titles.

This paper is concerned with the analysis and numerical solution of an $\boldsymbol {H}({\textbf {curl}})$-elliptic hemivariational inequality (HVI). One source of the HVI is through a temporal semidiscretization of a related hyperbolic Maxwell equation problem. An equivalent minimization principle is introduced, and the solution existence and uniqueness of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI are proved. Numerical analysis of the HVI is provided with a general Galerkin approximation, including a Céa’s inequality for convergence and error estimation. When the linear edge finite element method is employed, an optimal-order error estimate is derived under a suitable solution regularity assumption. A fully discrete scheme based on the backward Euler difference in time and a mixed finite element method in space is also analyzed, and stability estimates are derived for first-order terms of the fully discrete solution. Numerical results are reported on linear edge finite element solutions of the $\boldsymbol {H}({\textbf {curl}})$-elliptic HVI for numerical evidence of the theoretically predicted convergence order.

This paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

The diffusive-viscous wave equation arises in a variety of applications in geophysics, and it plays an important role in seismic exploration. In this paper, semi-discrete and fully discrete numerical methods are introduced to solve a general initial-boundary value problem of the diffusive-viscous wave equation. The spatial discretization is carried out through the finite element method, whereas the time derivatives are approximated by finite differences. Optimal order error estimates are derived for the numerical methods. Numerical results on a test problem are reported to illustrate the numerical convergence orders.

Aims: Protection of cardiac function following myocardial infarction was largely enhanced by bradykinin-pretreated cardiac-specific c-kit+ (BK-c-kit+) cells, even without significant engraftment, indicating that paracrine actions of BK-c-kit+ cells play a pivotal role in angiogenesis. Nevertheless, the active components of the paracrine actions of BK-c-kit+ cells and the underlying mechanisms remain unknown. This study aimed to define the active components of exosomes from BK-c-kit+ cells and elucidate their underlying protective mechanisms. Methods and results: Matrigel tube formation assay, cell cycle, and mobility in human umbilical vein endothelial cells (HUVECs) and hindlimb ischemia (HLI) in mice were applied to determine the angiogenic effect of condition medium (CM) and exosomes. Proteome profiler, microRNA sponge, Due-luciferase assay, microRNA-sequencing, qRT-PCR, and Western blot were used to determine the underlying mechanism of the angiogenic effect of exosomes from BK-c-kit+. As a result, BK-c-kit+ CM and exosomes promoted tube formation in HUVECs and the repair of HLI in mice. Angiogenesis-related proteomic profiling and microRNA sequencing revealed highly enriched miR-3059-5p as a key angiogenic component of BK-c-kit+ exosomes. Meanwhile, loss- and gain-of-function experiments revealed that the promotion of angiogenesis by miR-3059-5p was mainly through suppression of TNFSF15 (VEGI)-inhibited effects on vascular tube formation, cell proliferation and cell migration. Moreover, enhanced angiogenesis of miR-3059-5p-inhibited TNFSF15 has been associated with Akt/Erk1/2/Smad2/3-modulated signaling pathway. Conclusion: Our results demonstrated a novel finding that BK-c-kit+ cells enrich exosomal miR-3059-5p to suppress TNFSF15 and promote angiogenesis against hindlimb ischemia in mice.

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results., 33 pages,6 figures

In this paper, we provide stability analysis for a stationary Stokes hemivariational inequality where along the tangential direction of the slip boundary, an inclusion relation involving the generalized subdifferential of a superpotential is specified. With viscous incompressible fluid flows as application background, stability is analyzed for solutions with respect to perturbations in the superpotential and the density of external forces. We also present a result on the existence of a solution to an optimal control problem for the stationary Stokes hemivariational inequality.

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

In this paper, we consider the numerical solution of an evolutionary variational–hemivariational inequality arising in a dynamic contact problem. The material is assumed to be viscoelastic with short memory. The contact is featured by a normal damped response in the normal direction and by the Tresca friction law in the tangential direction. The linear finite elements are used to discretize the spatial variable. Optimal order error estimates are derived for the discrete velocity and discrete displacement under suitable solution regularity assumptions.

In this paper numerical approximation of history-dependent hemivariational inequalities with constraint is considered, and corresponding Cea’s type inequality is derived for error estimate. For a viscoelastic contact problem with normal penetration, an optimal order error estimate is obtained for the linear element method. A numerical experiment for the contact problem is reported which provides numerical evidence of the convergence order predicted by the theoretical analysis.

This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions.

In this paper, we study a family of discontinuous Galerkin (DG) fully discrete schemes for solving the second-order wave equation. The spatial variable discretization is based on an application of the DG method. The temporal variable discretization depends on a parameter $$\theta \in [0,1]$$ . Under suitable regularity hypotheses on the solution, optimal order error bounds are shown for the numerical schemes with $$\theta \in [\frac{1}{2},1]$$ , unconditionally with respect to the spatial mesh-size and the time-step, and for the numerical schemes with $$\theta \in [0,\frac{1}{2})$$ where a Courant–Friedrichs–Lewy stability condition is satisfied relating the mesh-size and the time-step. The optimal order error estimates are derived for $$H^{1}(\Omega )$$ and $$L^{2}(\Omega )$$ norms. Simulation results are reported to provide numerical evidence of the optimal convergence orders predicted by the theory.

In this paper, an equivalent minimization principle is established for a hemivariational inequality of the stationary Stokes equations with a nonlinear slip boundary condition. Under certain assumptions on the data, it is shown that there is a unique minimizer of the minimization problem, and furthermore, the mixed formulation of the Stokes hemivariational inequality has a unique solution. The proof of the result is based on basic knowledge of convex minimization. For comparison, in the existing literature, the solution existence and uniqueness result for the Stokes hemivariational inequality is proved through the notion of pseudomonotonicity and an application of an abstract surjectivity result for pseudomonotone operators, in which an additional linear growth condition is required on the subdifferential of a super-potential in the nonlinear slip boundary condition.

In this paper, we provide an alternative approach to establish the solution existence and uniqueness for elliptic variational-hemivariational inequalities. The new approach is based on elementary results from functional analysis, and thus removes the need of the notion of pseudomonotonicity and the dependence on surjectivity results for pseudomonotone operators. This makes the theory of elliptic variational-hemivariational inequalities more accessible to applied mathematicians and engineers. In addition, equivalent minimization principles are further explored for particular elliptic variational-hemivariational inequalities. Representative examples from contact mechanics are discussed to illustrate application of the theoretical results.

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.

In this paper, an electro-elastic frictional contact problem is studied numerically as a hemivariational inequality. Convergence of the Galerkin approximation for the hemivariational inequality is proved, and Cea’s type inequalities are derived for error estimation. The results are applied to the electro-elastic contact problem, and an optimal order error estimate is deduced for linear element approximation. Finally, two numerical examples are reported, providing numerical evidence of the optimal convergence order theoretically predicted.

Oxidative stress has a considerable influence on endothelial cell dysfunction and atherosclerosis. Acacetin, an anti‐inflammatory and antiarrhythmic, is frequently used in the treatment of myocarditis, albeit its role in managing atherosclerosis is currently unclear. Thus, we evaluated the regulatory effects of acacetin in maintaining endothelial cell function and further investigated whether the flavonoid could attenuate atherosclerosis in apolipoprotein E deficiency (apoE−/−) mice. Different concentrations of acacetin were tested on EA.hy926 cells, either induced or non‐induced by human oxidized low‐density lipoprotein (oxLDL), to clarify its influence on cell viability, cellular reactive oxidative stress (ROS) level, apoptotic ratios and other regulatory effects. In vivo, apoE−/− mice were fed either a Western diet or a chow diet. Acacetin pro‐drug (15 mg/kg) was injected subcutaneously two times a day for 12 weeks. The effects of acacetin on the atherosclerotic process, plasma inflammatory factors and lipid metabolism were also investigated. Acacetin significantly increased EA.hy926 cell viability by reducing the ratios of apoptotic and necrotic cells at 3 μmol/L. Moreover, 3 μmol/L acacetin clearly decreased ROS levels and enhanced reductase protein expression through MsrA and Nrf2 pathway through phosphorylation of Nrf2 and degradation of Keap1. In vivo, acacetin treatment remarkably attenuated atherosclerosis by increasing reductase levels in circulation and aortic roots, decreasing plasma inflammatory factor levels as well as accelerating lipid metabolism in Western diet‐fed apoE−/− mice. Our findings demonstrate the anti‐oxidative and anti‐atherosclerotic effects of acacetin, in turn suggesting its potential therapeutic value in atherosclerotic‐related cardiovascular diseases (CVD).

Background : Oxidative stress has a considerable influence on endothelial cell dysfunction and atherosclerosis. Acacetin, an anti-inflammatory and antiarrhythmic, is frequently used in the treatment of myocarditis, albeit its role in managing atherosclerosis is currently unclear. Thus, we evaluated the regulatory effects of acacetin in maintaining endothelial cell function and further investigated whether the flavonoid could attenuate atherosclerosis in apolipoprotein E deficiency (apoE -/- ) mice. Methods : Different concentrations of acacetin were tested on EA.hy926 cells, either induced or non-induced by human oxidized low density lipoprotein (oxLDL), to clarify its influence on cell viability, cellular reactive oxidative stress (ROS) level, apoptotic ratios and other regulatory effects. In vivo, apoE -/- mice were fed either a western diet or a chow diet. Acacetin prodrug (15mg/kg) was injected subcutaneously two times a day for 12weeks. The effects of acacetin on the atherosclerotic process, plasma inflammatory factors and lipid metabolism were also investigated. Results : Acacetin significantly increased EA.hy926 cell viability by reducing the ratios of apoptotic and necrotic cells at 3μM. Moreover, 3μM Acacetin clearly decreased ROS levels and enhanced reductase protein expression through MsrA and Nrf2 pathway through phosphorylation of Nrf2 and degradation of Keap1. In vivo , acacetin treatment remarkably attenuated atherosclerosis by increasing reductase levels in circulation and aortic roots, decreasing plasma inflammatory factor levels as well as accelerating lipid metabolism in Western diet-fed apoE -/- mice. Conclusions : Our findings demonstrate the anti-oxidative and anti-atherosclerotic effects of acacetin, in turn suggesting its potential therapeutic value in atherosclerotic-related cardiovascular diseases (CVD).

The energy-dependent form of the radiative transfer equation (RTE) is important in a variety of applications. In a previous paper the well-posedness and an energy discretization for the energy-dependent RTE were investigated. In this paper the fully discrete scheme is introduced and analysed. Optimal-order error estimates and a well-posedness analysis of the discrete system are provided. The theoretical results are validated through numerical examples.

The radiative transfer equation (RTE) arises in a wide variety of applications. In certain situations, the energy dependence is not negligible. In a series of two papers, we study the energy dependent RTE. In this first paper of the series, we focus on the well-posedness analysis and energy discretization. We use a mixed formulation so that the analysis covers both cases of non-vanishing absorption and vanishing absorption. We introduce a natural energy discretization scheme and derive an optimal order error estimate for the scheme. Angular discretization, spatial discretization and fully discrete schemes, as well as numerical simulation results, are the topics of the sequel.

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics.

The radiative transfer equation (RTE) arises in a wide variety of applications, in particular, in biomedical imaging applications associated with the propagation of light through the biological tissue. However, highly forward-peaked scattering feature in a biological medium makes it very challenging to numerically solve the RTE problem accurately. One idea to overcome the difficulty associated with the highly forward-peaked scattering is through the use of a delta-Eddington phase function. This paper is devoted to an RTE framework with a family of delta-Eddington-type phase functions. Significance in biomedical imaging applications of the RTE with delta-Eddington-type phase functions are explained. Mathematical studies of the problems include solution existence, uniqueness, and continuous dependence on the problem data: the inflow boundary value, the source function, the absorption coefficient, and the scattering coefficient. Numerical results are presented to show that employing a delta-Eddington-type phase function with properly chosen parameters provides accurate simulation results for light propagation within highly forward-peaked scattering media.

Optical tomography is an emerging and important molecular imaging modality. The aim of optical tomography is to reconstruct optical properties of human tissues. In this paper, we focus on reconstructing the absorption coefficient based on the radiative transfer equation (RTE). It is an ill-posed parameter identification problem. Regularization methods have been broadly applied to reconstruct the optical coefficients, such as the total variation (TV) regularization and the L1 regularization. In order to better reconstruct the piecewise constant and sparse coefficient distributions, TV and L1 norms are combined as the regularization. The forward problem is discretized with the discontinuous Galerkin method on the spatial space and the finite element method on the angular space. The minimization problem is solved by a Jacobian-based Levenberg-Marquardt type method which is equipped with a split Bregman algorithms for the L1 regularization. We use the adjoint method to compute the Jacobian matrix which dramatically improves the computation efficiency. By comparing with the other imaging reconstruction methods based on TV and L1 regularizations, the simulation results show the validity and efficiency of the proposed method.

This paper is devoted to the study of a hemivariational inequality modeling the quasistatic bilateral frictional contact between a viscoelastic body and a rigid foundation. The damage effect is built into the model through a parabolic differential inclusion for the damage function. A solution existence and uniqueness result is presented. A fully discrete scheme is introduced with the time derivative of the damage function approximated by the backward finite different and the spatial derivatives approximated by finite elements. An optimal order error estimate is derived for the fully discrete scheme when linear elements are used for the velocity and displacement variables, and piecewise constants are used for the damage function. Simulation results on numerical examples are reported illustrating the performance of the fully discrete scheme and the theoretically predicted convergence orders., Comment: 22 pages, 5 figures

This paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems, including variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. The constraint is relaxed by a penalty formulation and is re-stored as the penalty parameter tends to zero. The main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh-size and the penalty parameter approach zero simultaneously but independently. The convergence of the penalty based numerical methods is first established for a general elliptic variational-hemivariational inequality with constraints, and then for hemivariational inequalities and variational inequalities as special cases. Applications to problems in contact mechanics are described.

This paper represents a sequel to the previous one, where numerical solution of a quasistatic contact problem is considered for an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. Some preliminary error analysis for a fully discrete approximation of the contact problem was provided in the previous paper. In this paper, we consider a more general fully discrete numerical scheme for the contact problem, derive optimal order error bounds and present computer simulation results showing that the numerical convergence orders match the theoretical predictions., Comment: 13 pages, 6 figures

The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.

This paper presents a fast and optimal multigrid solver for the radiative transfer equation. A discrete-ordinate discontinuous-streamline diffusion method is employed to discretize the radiative transfer equation. Instead of utilizing conventional multigrid methods for spatial variables only, a spatial cascadic multigrid method and a full cascadic multigrid method are developed to achieve rapid convergence in iterative calculation. Preliminary analysis is also conducted, suggesting the optimal convergence rate. Numerical tests show a significant reduction of the computational time compared to conventional iterative methods for the radiative transfer equation.

A time-dependent Stokes fluid flow problem is studied with nonlinear boundary conditions described by the Clarke subdifferential. We present equivalent weak formulations of the problem, one of them in the form of a hemivariational inequality. The existence of a solution is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Finally, we present a result on the existence of a solution to an optimal control problem for the hemivariational inequality.

The non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 ( α ⩾ β ⩾ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 regularized function can be computed by applying the ST-(αℓ 1 − βℓ 2) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ 1 sparsity regularization. In this paper, we present two accelerated alternatives to the ST-(αℓ 1 − βℓ 2) algorithm by extending the PG method to the non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ 1-ball constraint by Morozov’s discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.

Variational–hemivariational inequalities and hemivariational inequalities form a powerful mathematical tool in modeling and studying problems in science and engineering where non-smooth, non-monotone and multi-valued relations among different physical quantities are present. In this paper, we consider the numerical solution of optimal control problems for variational–hemivariational inequalities or hemivariational inequalities, and prove the convergence of numerical solutions under rather general assumptions.

In this work, a residual type a posteriori error estimator is presented for finite element approximations of a variational inequality arising in hydrodynamics, which is governed by the stationary Stokes equations with a nonlinear slip boundary condition of friction type. Reliability of the estimator is rigorously proved, and efficient error control is analyzed, both are based on an equality problem with a Lagrangian multiplier. And an algorithm of recovering the multiplier is also provided. Numerical results are reported to illustrate the good performance of the estimator in the adaptive solution of the variational inequality, as well as the availability of the multiplier.

The diffusive-viscous wave equation arises in a variety of applications in geophysics such as the attenuation of seismic waves propagating in fluid-saturated solids, and frequency-dependent phenomena in porous media. This paper provides a well-posedness analysis on the initial-boundary value problem of the diffusive-viscous wave equation.

We consider the numerical solution of a variational inequality arising in the study of an incompressible Stokes flow with a nonlinear slip boundary condition of frictional type. We study several discontinuous Galerkin (DG) schemes for solving the problem, and derive reliable and efficient residual type a posteriori error estimators. An adaptive DG method is constructed based on the error estimators to solve the problem. Some numerical examples are presented to illustrate the efficiency of the adaptive DG method.

The paper is devoted to the derivation of a combined system of motion equations for solid and fluid in isotropic tight oil/gas sandstone media through volume averaging theorems (VAT). Based on the features of the media, four physical assumptions are proposed as the foundation for our derivation. More precisely, volume averaging theorems are applied to the micro-scale motion equations for both the solid and the fluid as well as to the stress-strain relations, resulting in a combined system of macro-scale equations for the tight oil/gas sandstone media. It is worth noting that the four assumptions may not be satisfied in the whole region. Nevertheless, since the characteristic diameter for applying VAT ranges between $10^{-6}$ meters and dozens of meters, we may split the entire domain into several sub-domains such that the four physical assumptions are satisfied in each sub-domain. By choosing a proper characteristic diameter of an averaging volume, we derive a formula for the fluid average pressure in terms of the divergence of the average displacement from the continuity equation of the fluid. As a result, the motion equations derived in this paper are simpler than the Biot equations, and are more suitable for inversion of porous medium parameters. When the fluid is gas and the compressional wave is considered, the derived motion equations can be simplified to the diffusive-viscous wave equation. Moreover, the explicit relationship between the coefficients in this equation and medium parameters is very important for gas detection in tight gas sandstone., 24 pages

Penalty methods approximate a constrained variational or hemivariational inequality problem through a sequence of unconstrained ones as the penalty parameter approaches zero. The methods are useful in the numerical solution of constrained problems, and they are also useful as a tool in proving solution existence of constrained problems. This paper is devoted to a theoretical analysis of penalty methods for a general class of variational–hemivariational inequalities with history-dependent operators. Unique solvability of penalized problems is shown, as well as the convergence of their solutions to the solution of the original history-dependent variational–hemivariational inequality as the penalty parameter tends to zero. The convergence result proved here generalizes several existing convergence results of penalty methods. Finally, the theoretical results are applied to examples of history-dependent variational–hemivariational inequalities in mathematical models describing the quasistatic contact between a viscoelastic rod and a reactive foundation.