In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

In this paper, we investigate the residual-based a posteriori error estimates of two-grid weak Galerkin (WG) methods for second order semilinear elliptic partial differential equations (PDEs). First, we propose two different two-grid weak Galerkin methods for the model problem and then establish a posteriori error estimators of the two-grid weak Galerkin methods. Theoretical analysis is given to prove the reliability and efficiency of the error estimators. We mainly study the lowest order case of WG element (Pj(T),Pℓ(∂T),RTj(T)) with j=ℓ=0 [19]. Numerical experiments are provided to confirm the theoretical results.

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H-1-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid arid the solution of two symmetric arid positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h = H-2.

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.

In this paper, we study the rectified linear unit (ReLU) artificial neural network (ANN) for grid adaptation in finite element method, which is used for solving differential equations (DEs) with initial/boundary condition. Compared with the classical adaptive finite element method (AFEM), ReLU ANN based on finite element method can keep the number of grid-points constant but change their relative location. Our numerical experiments show that approximate solutions obtained from the classical finite element method by ReLU ANN are more accurate than those obtained by AFEM.

为处理方程的拟线性性质,采用基于扩展混合有限元的两层网格离散方法研究拟线性抛物型积分微分方程.相对于经典的两层网格算法,基于扩展混合有限元方法的两层网格算法包含2步.在粗网格上,求解基于显式欧拉格式的线性问题;在细网格上,通过将非线性项基于粗网格解进行Taylor展开,从而求解一个线性化的方程组.理论和数值结果显示:当粗细网格步长满足h=H2时,该离散方法具有最优的收敛阶.

In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that if the two mesh sizes satisfy h = H2, then the two-grid method achieves the same convergence property as the Raviart-Thomas mixed finite element method. Finally, we give a numerical example to verify the theoretical results.

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.

In this paper, we will investigate a multigrid algorithm for poroelasticity problem by a new finite element method with homogeneous boundary conditions in two dimensional space. We choose Nedelec edge element for the displacement variable and piecewise continuous polynomials for the pressure variable in the model problem. In constructing multigrid algorithm, a distributive Gauss-Seidel iteration method is applied. Numerical experiments shows that the finite element method achieves optimal convergence order and the multigrid algorithm is almost uniformly convergent to mesh size h and parameter delta t on regular meshes.

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H , h satisfy the relation h = H 2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.

In this work, we investigate a novel two-level discretization method for semilinear reaction-diffusion equations. Motivated by the two-grid method for nonlinear partial differential equations (PDEs) introduced by Xu [18] on physical space, our discretization method uses a two-grid finite element discretization method for semilinear partial differential equations on physical space and a two-level finite difference method for the corresponding time space. Specifically, we solve a semilinear equations on a coarse mesh T-H (Omega) (partition of domain W with mesh size H) with a large time step size Theta and a linearized equations on a fine mesh T-h (Omega) (partition of domain Omega with mesh size h) using smaller time step size theta. Both theoretical and numerical results show that when h = H-2, theta = Theta(2), the novel two-grid numerical solution achieves the same approximate accuracy as that for the original seminlinear problem directly by finite element method with T-h (Omega) and theta.

In this work, we will investigate the numerical method we proposed in 1 and study the general convergence property of the method. We prove that for the model problem we studied, when its solution is analytic to each random variable in a bounded complex domain, the proposed method is a convergent and efficient scheme for solving the model problem. Some numerical examples are implemented.

Abstract In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯H⁢(D){\mathcal{T}_{H}(D)} with small scale of sparse collocation points η⁢(L,N){\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯h⁢(D){\mathcal{T}_{h}(D)} using large scale of sparse collocation points η⁢(ℓ,N){\eta(\ell,N)} (where η⁢(L,N),η⁢(ℓ,N){\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L,ℓ{L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h≈H3,η⁢(ℓ,N)≈(η⁢(L,N))3{{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}}, the novel two-level discretization method achieves the same convergence accuracy in norm ∥⋅∥ℒρ2⁢(Γ)⊗ℒ2⁢(D){\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} (ℒρ2⁢(Γ){\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ2{\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯h⁢(D){\mathcal{T}_{h}(D)} and large scale collocation points η⁢(ℓ,N){\eta(\ell,N)} in random spaces.

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu (1994), our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh T H with a low level stochastic collocation (corresponding to the polynomial space P P ) and solve linearized equations on a fine mesh T h using high level stochastic collocation (corresponding to the polynomial space P p ). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with T h and P p . The two-level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients. Numerical experiments are provided to verify the theoretical results.

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h(1/2)). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O ( h 1/2 ). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013