In this paper, we propose mixed finite element methods for the Reissner-Mindlin Plate Problem by introducing the bending moment as an independent variable. We apply the finite element approximations of the stress field and the displacement field constructed for the elasticity problem by Hu (J Comp Math 33: 283–296, 2015), Hu and Zhang (arXiv:1406.7457, 2014) to solve the bending moment and the rotation for the Reissner-Mindlin Plate Problem. We propose two triples of finite element spaces to approximate the bending moment, the rotation, and the displacement. The feature of these methods is that they need neither reduction terms nor penalty terms. Then, we prove the well-posedness of the discrete problem and obtain the optimal estimates independent of the plate thickness. Finally, we present some numerical examples to demonstrate the theoretical results.

This paper investigates two different Leslie matrix solutions for the reduced biquaternion matrix equation $AXB+CXD=E$. Through the permutation matrices, the complex decomposition of reduced biquaternion matrices, and the Kronecker product, by leveraging the specific attributes of Leslie matrices, we transform the constrained reduced biquaternion matrix equation into an unconstrained form. Consequently, we derive the necessary and sufficient conditions for the existence of solutions in the form of Leslie matrices to the reduced biquaternion matrix equation $AXB+CXD=E$ and provide a general expression for such solutions. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.

This paper investigates the scattering of biharmonic waves by a one-dimensional periodic array of cavities embedded in an infinite elastic thin plate. The transparent boundary conditions (TBCs) are introduced to formulate the problem from an unbounded domain to a bounded one. The well-posedness of the associated variational problem is demonstrated utilizing the Fredholm alternative theorem. The perfectly matched layer (PML) method is employed to reformulate the original scattering problem, transforming it from an unbounded domain to a bounded one. The TBCs for the PML problem are deduced, and the well-posedness of its variational problem is established. Moreover, the exponential convergence is achieved between the solution of the PML problem and that of the original scattering problem.

In this paper, we investigate the numerical method for the two-dimensional time-fractional Zakharov-Kuznetsov (ZK) equation. By the method of order reduction, the model is first transformed into an equivalent system. A nonlinear difference scheme is then proposed to solve the equivalent model with $min\{2,r\alpha \}$-th order accuracy in time and second-order accuracy in space, where $\alpha \in (0,1)$ is the fractional order and the grading parameter $r⩾1$. The existence of the numerical solution is carefully studied by the renowned Browder fixed point theorem. With the help of the Grönwall inequality and some crucial skills, we analyze the unconditional stability and convergence of the proposed scheme based on the energy method. Finally, numerical experiments are given to illustrate the correctness of our theoretical analysis.

The randomized block Kaczmarz (RBK) method is a randomized orthogonal projection iterative approach, which plays an important role in solving large-scale linear systems. A key point of this type of method is to select working rows effectively during iterations. However, in most of the RBK-type methods, one has to scan all the rows of the coefficient matrix in advance to compute probabilities or paving, or to compute the residual vector of the linear system in each iteration to determine the working rows. These are unfavorable for big data problems. To cure these drawbacks, we propose a semi-randomized block Kaczmarz (SRBK) method with simple random sampling for large-scale linear systems in this paper. The convergence of the proposed method is established. Numerical experiments on some real-world and large-scale data sets show that the proposed method is often superior to many state-of-the-art RBK-type methods for large linear systems.

An improved SSOR-like (ISSOR-like) preconditioner is proposed for the non-Hermitian positive definite linear system with a dominant skew-Hermitian part. The upper and lower bounds on the real and imaginary parts of the eigenvalues of the ISSOR-like preconditioned matrix and the convergence property of the corresponding ISSOR-like iteration method are discussed in depth. Numerical experiments show that the ISSOR-like preconditioner can effectively accelerate preconditioned GMRES.

A stable and high-order accurate solver for linear and nonlinear parabolic equations is presented. An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solution is performed by the recently developed Hierarchical Poincaré-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time step ensures that the coefficient matrix in the implicit solution of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N\log N)$ for the solution in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of the stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of the efficiency of the method in one and two dimensions are presented.

We propose the modulus-based cascadic multigrid (MCMG) method and the modulus-based economical cascadic multigrid method for solving the quasi-variational inequalities problem. The modulus-based matrix splitting iterative method is adopted as a smoother, which can accelerate the convergence of the new methods. We also give the convergence analysis of these methods. Finally, some numerical experiments confirm the theoretical analysis and show that the new methods can achieve high efficiency and lower costs simultaneously.

A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011; Sun in Math Numer Sin 34(1): 1–24, 2012; Sun in J Numer Methods Comput Appl 42(2): 104–125, 2021; Sun in Math Numer Sin 44(4): 433–465, 2022; Sun in Sci China Math 53(6): 859–894, 2023; Sun et al. in Chin Ann Math Ser B 44(5): 735–752, 2023). Different from traditional preconditioning algorithm with the discrete matrix directly, our geometric pre-processing algorithm (GPA) algorithm is based on so-called intrinsic geometric invariance, i.e., commutativity between the stiff matrix A and the grid mesh matrix $G:AG=GA$. Thus, the large-scale system solvers can be replaced with a much smaller block-solver as a pretreatment. In this paper, we study a sole PDE and assume G satisfies a periodic condition $G^m=I,m<<\text{dim}(G)$. Four special cases have been studied in this paper: two-point ODE eigen-problem, Laplace eigen-problems over L-shaped region, square ring, and 3D hexahedron. Two conclusions that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” and “commutativity of grid mesh matrix and mass matrix is the essential condition for the GPA algorithm” have been obtained.

In this paper, we shall carry out the L${}^2$-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation. The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis. This extension looks easy for most cases; however, there are a few troubles with obtaining good stability results under a standard CFL condition, especially, for those ${\mathcal{Q}}^k$-elements with lower degree k as stated in the one-dimensional case. To overcome this difficulty, we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose. In addition, an optimal error estimate on ${\mathcal{Q}}^k$-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.