A two‐grid stabilized mixed finite element method for Darcy‐Forchheimer model

A two‐grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two‐dimensional Darcy‐Forchheimer model. We use the derivative of a smooth function, , to approximate the derivative of in constructing the two‐grid algorithm. The two‐grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two‐grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.     

Error analysis of mixed finite element method for Poisson‐Nernst‐Planck system

To improve the convergence rate in L² norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson‐Nernst‐Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time‐dependent ionic concentrations equations. Optimal error estimates in norm for the electrostatic potential, and in and norms for the ionic concentrations are attained. As a by‐product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017     

A finite element–finite volume discretization of convection‐diffusion‐reaction equations with nonhomogeneous mixedboundary conditions: Error estimates

We consider a time‐dependent and a steady linear convection‐diffusion‐reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin–Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element–finite volume method: the diffusion term is discretized by Crouzeix–Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. The ‐ and the ‐error in the unsteady case and the H¹‐error in the steady one are estimated against the data, in such a way that no parameter enters exponentially into the constants involved. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1591–1621, 2016     

Superconvergence and gradient recovery for a finite volume element method for solving convection‐diffusion equations

We study the superconvergence of the finite volume element (FVE) method for solving convection‐diffusion equations using bilinear trial functions. We first establish a superclose weak estimate for the bilinear form of FVE method. Based on this estimate, we obtain the H¹‐superconvergence result: . Then, we present a gradient recovery formula and prove that the recovery gradient possesses the ‐order superconvergence. Moreover, an asymptotically exact a posteriori error estimate is also given for the gradient error of FVE solution.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1152–1168, 2014     

A new nonconforming finite element with a conforming finite element approximation for a coupled continuum pipe‐flow/Darcy model in Karst aquifers

Numerical method is considered for a coupled continuum pipe‐flow/Darcy model describing flow in porous media with an embedded conduit pipe. A new nonconforming element is constructed to solve the Darcy equation on porous matrix. The existence and uniqueness of the approximation solution are deduced. Optimal error estimates are obtained in and norms. Some numerical examples show the accuracy and efficiency of the presented method. With the same number of nodal‐points and the same amount of computation, the results using the new nonconforming element are much better than those by both conforming element and Wilson nonconforming element on the same mesh. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 778–798, 2016     

Convergence analysis of a new mixed finite element method for Biot's consolidation model

In this article, we propose a mixed finite element method for the two‐dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a‐priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained‐specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014     

Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions

In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has memory requirement and computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations.     

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

In this article, we study fast discontinuous Galerkin finite element methods to solve a space‐time fractional diffusion‐wave equation. We introduce a piecewise‐constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix‐vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from required by the traditional methods to log , where is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017     

Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizon

In this article, we deal with a rigorous error analysis for the finite element solutions of the two‐dimensional Cahn–Hilliard equation with infinite time. The error estimates with respect to are proven for the fully discrete conforming piecewise linear element solution under Assumption (A1) on the initial value and Assumption (A2) on the discrete spectrum estimate in the finite element space. The analysis is based on sharp a‐priori estimates for the solutions, particularly reflecting their behavior as . Numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the fully discrete mixed finite element methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 742–762, 2017     

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids, and then study the expanded mixed finite element method applied to the initial boundary value problem for the resulting degenerate parabolic equation for pressure. The bounds for the solutions, time derivative, and gradient of solutions are established. Utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions, a priori error estimates for solution are obtained in ‐norm, ‐norm as well as for its gradient in ‐norm for all . Optimal ‐error estimates are shown for solutions under some additional regularity assumptions. Numerical results using the lowest order Raviart–Thomas mixed element confirm the theoretical analysis regarding convergence rates. Published 2015. Numer Methods Partial Differential Eq 32: 60–85, 2016     

Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

In this article, a fourth‐order compact and conservative scheme is proposed for solving the nonlinear Klein‐Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge‐Kutta‐Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete ‐norm. Numerical results show that the integral method with variational limit gives an efficient fourth‐order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein‐Gordon equation compared with other methods under the same condition. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1283–1304, 2017     

Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017     

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.     

A mixed hp FEM for the approximation of fourth‐order singularly perturbed problems on smooth domains

We consider fourth‐order singularly perturbed problems posed on smooth domains and the approximation of their solution by a mixed Finite Element Method on the so‐called Spectral Boundary Layer Mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at an exponential rate when the error is measured in the energy norm. Numerical examples illustrate our theoretical findings.     

Three‐dimensional quadratic nonconforming brick element

A new nonconforming brick element with quadratic convergence for the energy norm is introduced. The nonconforming element consists of on a cube [?1,1]³, and 14 degree of freedom (DOF). Two types of DOF are introduced. One consists of the value at the eight vertices and six face‐centroids and the other consists of the value at the eight vertices and the integration value of six faces. Error estimates of optimal order are derived in both broken energy and norms for second‐order elliptic problems. If a genuine hexahedron, which is not a parallelepiped, is included in the partition, the proposed element is also convergent, but with a lower order. Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 158–174, 2014     

Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

In this article, we develop several first order fully discrete Galerkin finite element schemes for the Oldroyd model and establish the corresponding stability results for these numerical schemes with smooth and nonsmooth initial data. The stable mixed finite element method is used to the spatial discretization, and the temporal treatments of the spatial discrete Oldroyd model include the first order implicit, semi‐implicit, implicit/explicit, and explicit schemes. The ‐stability results of the different numerical schemes are provided, where the first‐order implicit and semi‐implicit schemes are the ‐unconditional stable, the implicit/explicit scheme is the ‐almost unconditional stable, and the first order explicit scheme is the ‐conditional stable. Finally, some numerical investigations of the ‐stability results of the considered numerical schemes for the Oldroyd model are provided to verify the established theoretical findings.     

Weak Galerkin finite element methods for linear parabolic integro‐differential equations

The semidiscrete and fully discrete weak Galerkin finite element schemes for the linear parabolic integro‐differential equations are proposed. Optimal order error estimates are established for the corresponding numerical approximations in both and norms. Numerical experiments illustrating the error behaviors are provided.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1357–1377, 2016     

Two‐grid method for two‐dimensional nonlinear Schrödinger equation by finite element method

A conservative two‐grid finite element scheme is presented for the two‐dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse‐grid solution as the initial guess. Moreover, error estimates are conducted for the two‐grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy in the two‐grid method. The numerical results show that this method is very effective.