This paper is concerned with the inverse scattering problems of simultaneously determining the unknown potential and unknown source for the biharmonic wave equation. We first derive an increasing stability estimate for the inverse potential scattering problem without a priori knowledge of the source function by multi-frequency active boundary measurements. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the potential function, where the latter decreases as the upper bound of the frequency increases. The key ingredients in the analysis are employing scattering theory to derive an analytic domain and resolvent estimates and an application of the quantitative analytic continuation principle. Utilizing the derived stability for the inverse potential scattering, we further prove an increasing stability estimate for the inverse source problem. The main novelty of this paper is that both the source and potential functions are unknown.

This paper considers a chemotaxis model coupled a stochastic Navier-Stokes equation in two-dimensional case. In non-convex bounded domain, it is proved that the stochastic chemotaxis-Navier-Stokes system possesses at least one global martingale weak solution when the chemotactic sensitivity function $\chi$ is nonsingular. In convex bounded domain, under the conditions of $\chi$ and per capita oxygen consumption rate $h$ are appropriately relaxed (where $\chi$ allows singularity), it is proved that the system admits a unique global mild solution. Our results generalize previously known ones.

A difference mixed finite element method based on the finite element pair $((P_0×P_1),$$(P_0×P_1),$$(P_1×P_0))×(P_1×P_1)$ for the three dimensional Poisson equation is studied. The method combines a mixed finite element discretization based on $(P_0 , P_0 , P_1 )×P_1$-element in $(x,y)$-plane and a finite difference discretization based on $(P_1,P_1,P_0)×P_1$-element in $z$-direction. This allows to transmit the finite element solution of the three dimensional Poisson equation in the direction $(x, y, z)$ into a series of finite element solution of the two dimensional Poisson equation in the direction $(x, y).$ Moreover, error estimates for the discrete approximation are derived. Numerical tests show the accuracy and efficiency for the method proposed.

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics, which yields an integro-differential equation that does not involve spatial derivatives of the displacement field. Therefore, it is particularly suitable for the description of cracks and their evolution in materials. Due to the nonlocal property of PD models, classical numerical methods usually generate dense stiffness matrices which require $\mathscr{O}(N^3)$ computational work and $\mathscr{O}(N^2)$ memory, where $N$ is the number of spatial unknowns. In this paper, we develop a fast collocation method for a bond-based linear viscoelastic peridynamic model in two space dimensions by exploring the structure of the stiffness matrix. The method has a computational work count of $\mathscr{O}(N {\rm log}N)$ per Krylov subspace iteration and $\mathscr{O}(N)$ memory requirements. In our model, the peridynamic neighborhood is a two-dimensional circular domain, so that one has to deal with the intersection of such circles and rectangular meshes used in the collocation method. We employ an inscribed polygon to approximate the peridynamic neighborhood. This improves the numerical accuracy of solution and reduces programming complexity. Numerical results show the utility of the proposed method.

A postprocessing-based a posteriori error estimates for the spectral Galerkin approximation of one-dimensional fourth-order boundary value problems are developed. Our approach begins by introducing a novel postprocessing technique aimed at enhancing the accuracy of the spectral Galerkin approximation. We prove that this post-processing step improves the convergence rate in both $L^2$- and $H^2$-norm. Using post-processed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the polynomial degree increases. We further extend the postprocessing technique and error estimators to more general one-dimensional even-order equations and to multidimensional fourth-order equations. The results of numerical experiments illustrate the efficiency of the error estimators.

In this study, we construct a relaxed second-order time-accurate scheme for the binary phase-field crystal model based on the recent paper [J. Wu, J. Yang and Z. Tan, Eng. Comput. (2023)]. This scheme can significantly improve the consistency between original and modified energies. Using the exponential SAV method, we construct the first- and second-order time-accurate schemes by using the backward Euler formula and the second-order backward difference formula, respectively. Then a new second-order numerical scheme with relaxation is developed. This new numerical scheme satisfies the energy dissipation law. Meanwhile, it is fully decoupled and efficient. Extensive numerical experiments are carried out in 2D and 3D to verify the accuracy and stability of the proposed scheme, as well as its ability to improve the consistency between the original energy and modified energy.

The momentum-conserving staggered-grid (MCS) scheme for two-layer shallow flow is revisited here. In the previous contribution, the MCS scheme has been successfully used to simulate various steady interfaces of two-layer shallow flow in channels of varying depth and width. In this contribution, the MCS scheme is extended to accommodate a time-dependent barotropic forcing, the barotropic tidal wave is assumed to be a harmonic function. In the MCS scheme, this barotropic forcing is assessed by applying suitable boundary conditions, and its presence would affect the exchange flow. The average exchange transport was then calculated for the barotropic forces with a range of amplitudes and periods, and the results obtained show good agreement with the analytical rigid-lid theory. Next, the MCS scheme is used to study the exchange flow in the Lombok Strait. Taking into account the geographical conditions of the Lombok Strait, a steady interface was simulated, which turned out to be the case of a maximal exchange flow with two controls at Karangasem Narrows (KN) and Nusa Penida Sill (NPS). Next, due to tidal waves in the period of April-May, the average exchange flow in Lombok Strait is estimated. Our findings agree well with the existing results.

An accurate, efficient and robust troubled-cell indicator (TCI) is always required by the discontinuous Galerkin methods for solving the nonlinear system of hyperbolic conservation laws. In this paper, we propose a new framework of TCI on general meshes using simple statistical analysis techniques which is easy to implement and efficient to run. In the framework troubled-cell indication is performed in each local stencil. We first pick the local stencils that contain the troubled cells via the range of the normalized troubled-cell indication values, and then detect the troubled cells in each picked stencil by a simple separation approach. The new TCI framework is implemented on several types of meshes with three indication variables. Numerical experiments conducted through simulating several classical shocked flows demonstrate that the new framework can capture the locations of shock waves accurately and lead to essentially nonoscillatory solutions. The results also verify the robustness of the framework.

This paper proposes difference finite element (DFE) methods for the Poisson equation in a four-dimensional (4D) region $ω × (0, L_4 ).$ The method converts the Poisson equation in a 4D region into a series of three-dimensional (3D) subproblems by the finite difference discretization in $(0, L_4)$ and deals with the 3D subproblems by the finite element discretization in $ω.$ In performing the finite element discretization, we select different discretization elements in the region $ω:$ hexahedral, pentahedral, and tetrahedral elements. Moreover, we prove the stability of the DFE solution $u_h$ and deduce the first-order convergence of $u_h$ with respect to the exact solution $u$ under $H^1$-error. Finally, three numerical examples are given to verify the accuracy and effectiveness of the DFE method.