Pattern formations by Gierer-Meinhardt (GM) activator-inhibitor model are considered in this paper. By linear analysis, critical value of bifurcation parameter can be evaluated to ensure Turing instability. Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization. We numerically show the convergence of our algorithm. Pattern transitions in irregular domains are shown. We also provide various parameter settings on some irregular domains for different patterns appeared in nature. To further simulate patterns in reality, we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.

In this article, we derive the a posteriori error estimators for a class of steady-state Poisson-Nernst-Planck equations. Using the gradient recovery operator, the upper and lower bounds of the a posteriori error estimators are established both for the electrostatic potential and concentrations. It is shown by theory and numerical experiments that the error estimators are reliable and the associated adaptive computation is efficient for the steady-state PNP systems.

This paper presents an absorbing boundary conditions (ABCs) for wave propagations on arbitrary computational domains. The purpose of ABCs is to eliminate the unwanted spurious reflection at the artificial boundaries and minimize the finite size effect. Traditional methods are usually complicate in theoretical derivation and implementation and work only for very limited types of boundary geometry. In contrast to other existing methods, our emphasis is placed on the ease of implementation. In particular, we propose a method for which the implementation can be done by fitting or learning from the simulation data in a larger domain, and it is insensitive to the geometry and space dimension of the computational domain. Furthermore, a stability criterion is imposed to ensure the stability of the proposed ABC. Numerical results are presented to demonstrate the effectiveness of our method.

In this work, we present a high-order discontinuous Galerkin method for the shallow water equations incorporating horizontal temperature gradients (also known as the Ripa model), which exactly maintains the lake at rest steady state. Herein, we propose original numerical fluxes defined on the basis of the hydrostatic reconstruction idea and a simple source term approximation. This novel approach allows us to achieve the well-balancing of the discontinuous Galerkin method without complication. Moreover, the proposed method retains genuinely high-order accuracy for smooth solutions and it shows good resolution for discontinuous solutions at the same time. Rigorous numerical analysis as well as extensive numerical results all verify the good performances of the proposed method.

For the low-order finite element pair $P_1-P_1$, based on full domain partition technique, a parallel pressure projection stabilized finite element algorithm for the Stokes equation with nonlinear slip boundary conditions is designed and analyzed. From the definition of the subdifferential, the variational formulation of this equation is the variational inequality problem of the second kind. Each subproblem is a global problem on the composite grid, which is easy to program and implement. The optimal error estimates of the approximate solutions are obtained by theoretical analysis since the appropriate stabilization parameter is chosen. Finally, some numerical results are given to demonstrate the high efficiency of the parallel stabilized finite element algorithm.

In this paper, we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations (BSPDEs). By using finite element methods in the physical space domain and the Euler method in the time domain, we propose a spatial finite element semi-discrete scheme and a spatio-temporal full discrete scheme for solving the BSPDEs. Errors of the schemes are rigorously analyzed and theoretical error estimates with convergence rates are obtained.

In this article, a Crank-Nicolson/Explicit scheme is designed and analyzed for the time-dependent natural convection problem with nonsmooth initial data. The Galerkin finite element method (FEM) with stable MINI element is used for the velocity and pressure and linear polynomial for the temperature. The time discretization is based on the Crank-Nicolson scheme. In order to simplify the computations, the nonlinear terms are treated by the explicit scheme. The advantages of our numerical scheme can be listed as follows: (1) The original problem is split into two linear subproblems, these subproblems can be solved in each time level in parallel and the computational sizes are smaller than the origin one. (2) A constant coefficient linear discrete algebraic system is obtained in each subproblem and the computation becomes easy. The main contributions of this work are the stability and convergence results of numerical solutions with nonsmooth initial data. Finally, some numerical results are presented to verify the established theoretical results and show the performances of the developed numerical scheme.

In this paper, we study a new finite element method for poroelasticity problem with homogeneous boundary conditions. The finite element discretization method is based on a three-variable weak form with mixed finite element for the linear elasticity, i.e., the stress tensor, displacement and pressure are unknown variables in the weak form. For the linear elasticity formula, we use a conforming finite element proposed in [11] for the mixed form of the linear elasticity and piecewise continuous finite element for the pressure of the fluid flow. We will show that the newly proposed finite element method maintains optimal convergence order.

Dynamic transient responses of rotating twisted plate under the air-blast loading and step loading respectively considering the geometric nonlinear relationships are investigated using classical shallow shell theory. By applying energy principle, a novel high dimensional nonlinear dynamic system of the rotating cantilever twisted plate is derived for the first time. The use of variable mode functions by polynomial functions according to the twist angles and geometric of the plate makes it more accurate to describe the dynamic system than that using the classic cantilever beam functions and the free-free beam functions. The comparison researches are carried out between the present results and other literatures to validate present model, formulation and computer process. Equations of motion describing the transient high dimensional nonlinear dynamic response are reduced to a four degree of freedom dynamic system which expressed by out-plane displacement. The effects of twisted angle, stagger angle, rotation speed, load intensity and viscous damping on nonlinear dynamic transient responses of the twisted plate have been investigated. It's important to note that although the homogeneous and isotropic material is applied here, it might be helpful for laminated composite, functionally graded material as long as the equivalent material parameters are obtained.

This paper presents a new concept called Unified and Integrated Method for a shear deformable beam element. In this method, Timoshenko beam theory is unified and integrated in such a way that takes into account the effect of transverse shear and maintains the shear locking free condition at the same time to generate proper behavior in the analysis of thin to thick beams. The unified and integrated method is applied to finite element analysis (FEA) and isogeometric analysis (IGA) on two-node beam element. This method will be used to analyze uniformly loaded beams with various boundary conditions. A shear influence factor of $\phi$, which is a function of beam thickness ratio $(L/h)$, is expressed explicitly as control of the transverse shear strain effect. The analysis gives interesting results showing that applying the unified and integrated method in FEA and IGA will yield exact values of DOF's and displacement function even when using only a single element. Numerical examples demonstrate the validity and efficiency of the unified and integrated methods.