Kexin Li

This paper develops a fully discrete finite element method with first-order temporal accuracy and second-order spatial accuracy for a class of Navier–Stokes-type viscoelastic fluid equations with a tempered power-law memory kernel. The finite element method is employed for spatial discretization, while a first-order convolution quadrature is utilized to approximate the nonlocal memory term, and a semi-implicit scheme is adopted to treat the nonlinear convection term. The proposed method is rigorously analyzed in terms of stability, consistency, and convergence. In particular, stability estimates, truncation error bounds, and optimal L2-norm error estimates for the velocity are established. Extensive numerical experiments are conducted to verify the theoretical convergence rates for both manufactured solutions and benchmark problems without available analytical solutions. Furthermore, a two-dimensional 4:1 planar contraction channel is considered to investigate the impact of viscoelastic effects on complex flow dynamics. The numerical results reveal that the elastic stress coefficient and the tempering rate parameter play a crucial role in regulating the evolution of corner vortices and recirculation structures. In particular, the competition between viscous dissipation and elastic stress effects leads to non-monotonic vortex evolution, including vortex suppression and subsequent enlargement with increasing elasticity. These observations demonstrate the capability of the proposed method to capture the characteristic flow behaviors of viscoelastic fluids, including elasticity-induced instability and drag-reduction phenomena.

In this work, we propose a novel two-stage image segmentation method addressing the issue of intensity inhomogeneities in images. In the first stage, a non-convex variational decomposition model based on a tight wavelet frame is proposed. The proposed non-convex variational decomposition model is a weighted sum of three terms: the data fitting term, the non-convex regularization term based on the tight wavelet frame for the structural component (piecewise constant), and the second-order gradient regularization term for the intensity component (smooth). The existence of a solution for the proposed decomposition model is established, and the uniqueness is guaranteed under certain conditions. In the second stage, a thresholding technique is applied to the piecewise constant component obtained in the first stage to achieve the final image segmentation result. This two-stage strategy focuses on the structural features of the image during image segmentation, thereby mitigating the impact of intensity inhomogeneities. In addition, this paper develops an effective algorithm based on variational Bayesian inference that simultaneously segments images and selects model parameters, reducing the burden of manual parameter adjustment. Numerical experimental results demonstrate that the proposed method not only effectively reduces illumination and shadow interference but also exhibits significant competitiveness in processing images with intensity inhomogeneities.

This paper focuses on the numerical solution of the modified Euler–Bernoulli equation incorporating nonlinear nonlocal viscous damping, which is suitable for modeling vibration phenomena in beams and plates. This formulation is relevant for modeling vibrational phenomena in engineering structures. The proposed strategy produces an approximation of the unknown solution through a two‐step numerical process. First, the Galerkin finite element method is used to derive a spatially semi‐discrete scheme. Then, a time‐stepping difference method is applied to accomplish a fully discrete Galerkin scheme. The energy method is employed to establish the stability and uniqueness of both the semi‐discrete and fully discrete solutions. The main difficulty in this work lies in handling the nonlinear viscous damping term during discretization and analysis. Optimal error estimates are obtained for both discretizations. Numerical simulations confirm the theoretical predictions and indicate that the fully discrete scheme efficiently dissipates energy.

In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into an equivalent form via the convolution method. Two time discretization strategies are then applied to approximate each term in the transformed equation, yielding two fully discrete schemes based on a spatial compact finite difference method. To reduce computational cost, the alternating direction implicit (ADI) technique is employed. We prove that both ADI compact schemes are unconditionally stable and convergent. Under solution regularity, the first scheme achieves $α(0)$-order accuracy in time and fourth-order accuracy in space, while the second scheme attains second-order accuracy in time and fourth-order accuracy in space. Numerical experiments confirm the theoretical error estimates and demonstrate the efficiency of the proposed methods.