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The element-free Galerkin (EFG) method, which constructs shape functions via moving least squares (MLS) approximation, represents a fundamental and widely studied meshless method in numerical computation. Although it achieves high computational accuracy, the shape functions are more complex than those in the conventional finite element method (FEM), resulting in great computational requirements. Therefore, improving the computational efficiency of the EFG method represents an important research direction. This paper systematically reviews significant contributions from domestic and international scholars in advancing the EFG method. Including the improved element-free Galerkin (IEFG) method, various interpolating EFG methods, four distinct complex variable EFG methods, and a series of dimension splitting meshless methods. In the numerical examples, the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steady-state anisotropic heat conduction, 3D elastoplasticity, and large deformation problems, as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.

The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.

In this paper, we study the Caputo–Hadamard fractional partial differential equation where the time derivative is the Caputo–Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.

In recent years, meshless methods have been increasingly applied to the simulation of various engineering problems due to their inherent advantages over traditional mesh-based approaches, including greater flexibility, independence from predefined meshing, simpler adaptive analysis, improved automation, and suitability for complex problems. Several meshless methods have been used for porous media simulation, and are broadly categorized into collocation, global weak form and local weak form methods. In this study, a comprehensive comparison of the applicability of these three categories of meshless methods for simulating coupled flow and transport problems in porous media is presented. The Radial Point Collocation Method (RPCM) (strong form), the Element Free Galerkin Method (EFGM) (global weak form) and the Meshless Local Petrov Galerkin (MLPG) method (local weak form) are implemented and systematically compared. These methods are applied to the analysis of flow in a synthetic regular domain aquifer, flow and non-reactive contaminant transport in a synthetic irregular boundary porous media problem and groundwater flow in a field aquifer located in India. The simulated groundwater heads are compared with analytical solution, observed field data and results obtained from widely used MODFLOW-MT3DMS models. The deviation of the solutions from the analytical solution is in the range of 0.67% to 0.16% for the hypothetical case study. For the field-scale case study, mean absolute error of 0.183%, 0.181% and 0.188% are obtained for the RPCM, EFGM and MLPG models, respectively, outperforming MODFLOW, which exhibits a deviation of 0.254% from observed values. Overall, the present study reaffirms the practical applicability of these meshless methods for real-world groundwater problems and provides valuable insights into the utilization of each category of meshless method, with respect to problem type, computational efficiency and accuracy requirements.

This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge–Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step τ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent of the mesh-size h, even though the advection term is treated explicitly. The last is that under this time step condition, we obtain optimal error estimates in both space and time for the third order IMEX Runge–Kutta time-marching coupled with LDG spatial discretization. Numerical experiments are also given to verify the main results.

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the L ∞ -norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.

By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

The Keller-Segel (KS) chemotaxis equation is a widely studied mathematical model for understanding the collective behavior of cells in response to chemical gradients. This paper investigates the direct discontinuous Galerkin method with interface correction (DDGIC) for one-dimensional and two-dimensional KS equations governing the cell density and chemoattractant concentration. We establish error estimates for the proposed scheme under suitable smoothness assumptions of the exact solutions. Numerical experiments are conducted to validate the theoretical results. We explore the impact of different coefficient settings in the numerical fluxes on the error of the DDGIC method on uniform and nonuniform meshes. Our findings reveal that the DDGIC method achieves optimal convergence rates with any admissible coefficients for polynomials of odd degrees, while the accuracy of the cell density is sensitive to the numerical flux coefficient in the chemoattractant concentration for polynomials of even degrees. These results hold regardless of whether the mesh is uniform or nonuniform.