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The lowest-order weak Galerkin finite element method is introduced and analysed for second order elliptic interface problems with discontinuous coefficients and solutions. Compared with the existing weak Galerkin finite element methods for interface problems, this method has a simpler formulation and superconvergence property. Hence it is suitable for more complicated interfaces and domain configurations. We derive the superconvergence error estimates in discrete H 1 and L 2 norms. Furthermore, we verify the theoretical results by numerical experiments. Optimal convergence rates are achieved for examples with low-regularity solutions. [ABSTRACT FROM AUTHOR]

We develop a temporally second-order stabilizer-free weak Galerkin (SFWG) finite element method with unequal time steps for reaction-subdiffusion equation in multiple space dimensions. In consideration of initial singularity of the solution, we prove a sharp error estimate on nonuniform time steps by two tools: discrete fractional Grönwall inequality and error convolution structure analysis. Furthermore, the fully discrete scheme achieves second-order accuracy in time by employing the graded temporal partition. Several numerical experiments are included to confirm the sharpness of theoretical results. [ABSTRACT FROM AUTHOR]

We study a fully‐discrete stabilizer‐free weak Galerkin (SFWG) finite element method for the initial‐boundary value problem of variable‐order Caputo time‐fractional diffusion equation. Optimal convergence order in L2$$ {L}_2 $$ norm for L1$$ L1 $$ fully‐discrete SFWG scheme is obtained. Furthermore, we develop a fast evaluation algorithm to improve the computational efficiency. Numerical experiments are performed to verify theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider a weak Galerkin finite element method and a nonuniform two‐step backward differentiation formula scheme for solving the Allen–Cahn equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. It is well‐known that the Allen–Cahn equations have a nonlinear stability property, that is, the free‐energy functional decreases with respect to time. A corresponding energy stability analysis of the discretized system is developed. Optimal order of L2‐norm error estimates are derived. Numerical experiments are preformed to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a robust weak Galerkin discretization for the incompressible miscible displacement problem in porous media, which is a nonlinear time-dependent system. The scheme we constructed is positive definite and flexible by arbitrarily shaped polygonal mesh without imposing extra conditions on the stabilization. The other feature of our method is that there is no limitation on the ratio of time-step and spatial mesh size. Error estimates are derived for concentration and pressure in the L 2 norm and energy norm, respectively. Extensive numerical experiments, including realistic test cases, have been conducted to verify our theoretical results and the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k + 1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H 1 and standard L 2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings. • A special variational form possessing two built-in parameters is proposed for the one-dimensional nonlinear convection-diffusion problem. • The WG scheme, developed in this paper, is very flexible because different stencils can be obtained by tuning the built-in weighted parameters. • The optimal error estimates in both standard L 2 and discrete H 1 norms are derived for our numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L 2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various numerical examples are presented, showing that the method is as effective as the weak Galerkin method. [ABSTRACT FROM AUTHOR]

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study a weak Galerkin finite element method (WG-FEM) with second-order accuracy in time for solving one type of parabolic equations. We apply a discrete weak gradient operator on discontinuous piecewise polynomials in the numerical scheme. Fully-discrete WG finite element scheme is introduced and analyzed. Optimal order error estimates are derived in the L 2 norm. Numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We present some numerical schemes based on the weak Galerkin finite element method for one class of Sobolev equations, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. The proposed schemes will be proved to have good numerical stability and high order accuracy when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme. Finally, some numerical results are given to verify our analysis for the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study a stabilizer-free weak Galerkin (SFWG) finite element method with second-order accuracy in time for solving time-fractional diffusion equation. We apply the SFWG finite element method to discretize the Laplace operator and the L 2 - 1 σ formula for the Caputo fractional derivative. Optimal convergence orders of semi-discrete scheme in L 2 norm and H 1 norm and fully discrete L 2 - 1 σ -SFWG scheme in L 2 norm are obtained. Numerical experiments are performed to verify theoretical results. [ABSTRACT FROM AUTHOR]

A new conforming discontinuous Galerkin method, which is based on weak Galerkin finite element method, is introduced for solving second order elliptic interface problems with discontinuous coefficient. The numerical method studied in this paper has no stabilizer and fewer unknowns compared with the known weak Galerkin algorithms. The error estimates in H 1 and L 2 norms are established, which are the optimal order convergence. Numerical experiments demonstrate the performance of the method, confirm the theoretical results of accuracy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the solution of parabolic equation using the modified weak Galerkin finite element procedure, which is named as MWG-FEM, based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties, in which the classical gradient operator is replaced by a modified weak gradient operator. Optimal order error estimates in a discrete norm and norm are established for the corresponding modified weak Galerkin finite element solutions. Finally, we numerically verify the convergence theory for the MWG-FEM through some examples. [Copyright &y& Elsevier]

Abstract: In modern numerical simulation of prospecting and exploiting oil–gas resources and environmental science, it is important to consider a numerical method for nonlinear convection-dominated diffusion problems. Based on actual conditions, such as the three-dimensional characteristics of large-scale science-engineering computation, we present a kind of characteristic finite volume element method. Some techniques, such as calculus of variations, commutating operators, the theory of prior estimates and techniques, are adopted. Suboptimal order error estimate in norm and optimal order error estimate in norm are derived to determine the errors for the approximate solution. Numerical results are presented to verify the performance of the scheme. [Copyright &y& Elsevier]

Abstract: The upwind finite volume element method based on straight triangular prism partition is put forward for the nonlinear convection-diffusion problems. Some techniques, such as calculus of variations, commutating operator and prior estimates, are adopted. Optimal order error estimate in L 2 is derived to determine the error in approximate solution. Numerical experiment shows that the method is effective for avoiding numerical diffusion and nonphysical oscillations and reducing work time. [Copyright &y& Elsevier]

• We have provided comparison of CPU time for the standard WG-FEM and the fast WG-FEM in Table 4. • All references have been updated. • We have explained the super-convergence phenomena of | | ∇ w (e h) | | in the numerical experiments. • We have re-phased the "a new application of WG-FEM" according to reviewer's sugguestion. • The definition of Pk (K) has been corrected. • All minor mistakes pointed out by the reviewers have been corrected accordingly. • A detailed response to reviewers is attached to the revised manuscript. This paper is concerned with the application of a weak Galerkin finite element method (WG-FEM) to the time-fractional diffusion equation. The WG-FEM with L1-formula and the fast evaluation scheme based on WG-FEM are designed. The optimal convergence rates for both semi-discrete and fully discrete WG-FEM schemes are obtained, and the stability analysis for the semi-discrete WG-FEM is derived. Numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider a weak Galerkin finite element method (WG-FEM) for solving one type of viscoelastic wave equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. We introduce both the semi- and fully-discrete WG-FEMs and establish the corresponding stability estimates. Optimal order error estimate in H 1 -norm is derived. Numerical experiments are preformed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a SAV method for the Cahn–Hilliard type phase field model with new dynamic boundary conditions. By using the continuous finite element method in space and the backward difference method in time, the fully discrete numerical schemes preserving the energy law are constructed. Numerical examples show that the proposed scheme can simulate the phase field model well even in a relatively rough grid. [ABSTRACT FROM AUTHOR]

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k + 1 for the discrete weak derivative space. The optimal order error estimates in a discrete H 1 norm and the standard L 2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems. [ABSTRACT FROM AUTHOR]

Abstract We study weak Galerkin (WG) finite element method (FEM) for solving nonlinear convection-diffusion problems. A WG finite element scheme is presented based on a new variational form. We prove the energy conservation law and stability of the continuous time WG FEM. In particular, optimal order error estimates are established for the WG FEM approximation in both a discrete H 1-norm and L 2-norm. Numerical experiments are performed to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide an expanded mixed finite element method for a class of two-dimensional Sobolev equations. The optimal error estimates for a semi-discrete scheme and a fully discrete scheme are obtained. Also numerical examples are stated to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the multigrid algorithm for a quad-curl problem based on C 0 interior penalty method on bounded polygonal domains. It is shown that the contraction numbers of W -cycle multigrid algorithm for the C 0 interior penalty method converge uniformly with rate of m − α , where m is the number of pre-smoothing (and post-smoothing) steps, α is the index of elliptic regularity. The performance of the multigrid fast solver is illustrated by numerical results. [ABSTRACT FROM AUTHOR]

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise P k (k ≥ 1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [ P j (K) ] 2 (j ≥ j 0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L 2 projection operators, optimal error estimates of O (Δ t + h k + 1) in the L 2 norm and O (Δ t + h k) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis. • Optimal estimate of stabilizer-free WG method for drift-diffusion model is derived. • The method is flexible while reducing complexity in analyzing and programming. • A Ritz projection that can be applied to estimate other problems is introduced. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a meshless method named Moving Least Squares Particle Hydrodynamics (MLSPH) method is applied to obtain numerical solutions of the Burgers' equation with a set of initial and boundary conditions. The influence of kernel function and smoothing length on the accuracy of MLSPH method is investigated. Several various forms of one- and two- dimensional Burgers' equations are successfully calculated. It is concluded that the proposed method can obtain second-order accuracy. The way of non-uniform distribution is applied to present the desired performance near the shocks. The numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods. The satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a fuzzy bidirectional flow framework based on generalized fuzzy set is presented to sharpen image by reducing its edge width, which performs a fuzzy backward (inverse) diffusion along the gradient direction to the isophote line (edge), while does a certain forward diffusion along the tangent direction on the contrary. Gaussian smoothing to the second normal derivative of an image is used to decide its zero-crossing, which results in a robust sharpening process against noise. Controlled by the image gradient magnitude, the fuzzy membership function guarantees a natural transition across different areas. To preserve image features, the nonlinear diffusion coefficients are locally adjusted according to the directional derivatives of the image. Experiments on real images demonstrate that the algorithm substantially improves the visual quality of the enhanced image over some relevant equations. [Copyright &y& Elsevier]