Weimin Han

Cheng Wang, Qiwei Sheng, Weimin Han

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.

Mengwu Guo, Weimin Han, Hongzhi Zhong

A generalized constitutive relation error is proposed in an analogous form to Fenchel-Young inequality on the basis of the key idea of Legendre-Fenchel duality theory. The generalized constitutive relation error is linked with the global errors of some admissible solutions for the problem in question, and is of wide applicability, especially in a posteriori error estimations of numerical methods. A class of elliptic variational inequalities is examined using the proposed approach and a strict upper bound of global energy errors of admissible solutions is obtained.

Xiajie Huang, Fei Wang, Weimin Han et al.

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

Weimin Han, Shengda Zeng

This paper is devoted to the well-posedness analysis of a nonstationary Stokes hemivariational inequality for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. In a recent paper [19], well-posedness of the nonstationary Stokes hemivariational inequality is studied for both the velocity and pressure fields. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations for both the velocity and pressure fields. In this paper, a refined well-posedness analysis is provided on the nonstationary Stokes hemivariational inequality under more natural assumptions on the problem data. The solution existence is first shown for the velocity field through a limiting procedure based on temporally semi-discrete approximations of a reduced problem and then the pressure field is recovered with the help of an inf-sup property. In this way, assumptions on the source term and the initial velocity needed in [19] are weakened, and a compatibility condition on initial values of the data is dropped. Moreover, several hemivariational inequalities are introduced for the mathematical model and their equivalence is explored.

Chenhui Zhu, Fei Wang, Weimin Han

This paper presents a numerical approach to the stochastic obstacle problem using the stochastic Galerkin (SG) method. Due to the low regularity of the solution, linear finite elements are employed in both the physical and random variable spaces. Properties of random fields and variational inequalities of the first kind are employed to establish the well-posedness of the problem. Finite element spaces are introduced to construct suitable approximation subspaces, and a comprehensive SG formulation is proposed to solve the stochastic obstacle problem. Well-posedness of the discrete formulation is shown and an optimal error estimate for the numerical solution in the $H^1$-norm is derived. Numerical experiments validate the effectiveness of the SG method, showing that both the expectation error and second moment error converge at a rate of $O(h)$ in the $H^1$-norm, consistent with theoretical predictions.

Danfu Han, Weimin Han, Michal Jureczka et al.

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.