Adaptive finite element approximations on nonmatching grids for second‐order elliptic problems

In this article we consider a finite element approximation for a model elliptic problem of second order on non‐matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

In this article, we establish the existence and uniqueness of solutions to the coupled reaction–diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank–Nicolson and the predictor–corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second‐order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme.     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity‐pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. By using the weak Galerkin approach, we consider the two‐dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1352–1373, 2017     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

On time integration error estimation and adaptive time stepping in structural dynamics

In this paper we present two error estimators resp. indicators for the time integration in structural dynamics. Based on the equivalence between the standard Newmark scheme and a Galerkin formulation in time [1] for linear problems a global time integration error estimator based on duality [3] can also be derived for the Newmark scheme. This error estimator is compared to an error indicator based on a finite difference approach in time [2]. Finally an adaptive time stepping scheme using the global estimator and the local indicator is presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation

Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A posteriori error estimates for the mixed finite element method with Lagrange multipliers

We consider the mixed finite element method with Lagrange multipliers as applied to second‐order elliptic equations in divergence form with mixed boundary conditions. The corresponding Galerkin scheme is defined by using Raviart‐Thomas spaces. We develop a posteriori error analyses yielding a reliable and efficient estimate based on residuals, and a reliable and quasi‐efficient estimate based on local problems, respectively. Several numerical results illustrate the suitability of these a posteriori estimates for the adaptive computation of the discrete solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

Analysis of the HDG method for the stokes–darcy coupling

In this article, we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. We consider a fully‐mixed formulation in which the main unknowns in the fluid are given by the stress, the vorticity, the velocity, and the trace of the velocity, whereas the velocity, the pressure, and the trace of the pressure are the unknowns in the porous medium. In addition, a suitable enrichment of the finite dimensional subspace for the stress yields optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, similarly as in previous articles dealing with development of the a priori error estimates, we use the projection‐based error analysis to simplify the corresponding study. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 885–917, 2017     

Conservative flux recovery from the Q1 conforming finite element method on quadrilateral grids

Compared with standard Galerkin finite element methods, mixed methods for second‐order elliptic problems give readily available flux approximation, but in general at the expense of having to deal with a more complicated discrete system. This is especially true when conforming elements are involved. Hence it is advantageous to consider a direct method when finding fluxes is just a small part of the overall modeling processes. The purpose of this article is to introduce a direct method combining the standard Galerkin Q1 conforming method with a cheap local flux recovery formula. The approximate flux resides in the lowest order Raviart‐Thomas space and retains local conservation property at the cluster level. A cluster is made up of at most four quadrilaterals. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 104–127, 2004     

对流扩散方程一类改进的特征线修正有限元方法

１引言在地下水污染,地下渗流驱动,核污染,半导体等问题的数值模拟中,均涉及抛物型对流扩散方程（或方程组）的数值求解问题．这些对流扩散型偏微分方程（或方程组）具有共同的特点：对流的影响远大于扩散的影响,即对流占优性,对流占优性给问题的数值求解带来许多困难,因此对流占优问题的有效数值解法一直是计算数学中重要的研究内容．用通常的差分法或有限元法进行数值求解将出现数值振荡．为了克服数值振荡,提出各种迎风方法和修正的特征方法并在这些问题上得到成功的实际应用、８０年代,Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［２］等…     

Galerkin-FEM for obtaining the numerical solution of the linear fractional Klein-Gordon equation

In this paper, an efficient numerical method for solving the linear fractional Klein-Gordon equation (LFKGE) is introduced. The proposed method depends on the Galerkin finite element method (GFEM) using quadratic B-spline base functions and replaces the Caputo fractional derivative using $L2$ discretization formula. The introduced technique reduces LFKGE to a system of algebraic equations, which solved using conjugate gradient method. The study the stability analysis to the approximation obtained by the proposed scheme is given. To test the accuracy of the proposed method we evaluated the error norm $L_{2}$. It is shown that the presented scheme is unconditionally stable. Numerical example is given to show the validity and the accuracy of the introduced algorithm.     

Analysis of a new BEM‐FEM coupling for two‐dimensional fluid‐solid interaction

In this work we present a new numerical method, based on a coupling of finite and boundary elements, to solve a fluid‐solid interaction problem in the plane. The discrete method uses classical Lagrange finite elements adapted to curved boundaries for the field variable and spectral approximation of the unknowns on the artificial boundary. We provide error estimates for this Galerkin scheme and propose a full discretization based on elementary quadrature formulae, showing that the perturbation due to numerical integration preserves the optimal rate of convergence. We also suggest an iterative method to solve the complicated linear systems arising from this type of schemes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Stabilization method for continuous approximation of transient convection problem

The aim of this work is to study a new finite element (FE) formulation for the approximation of nonsteady convection equation. Our approximation scheme is based on the Streamline Upwind Petrov Galerkin (SUPG) method for space variable, x, and a modified of the Euler implicit method for time variable, t. The most interest for this scheme lies in its application to resolve by continuous (FE) method the complex of viscoelastic fluid flow obeying an Oldroyd‐B differential model; this constituted our aim motivation and allows us to treat the constitutive law equation, which expresses the relation between the stress tensor and the velocity gradient and includes tensorial transport term. To make the analysis of the method more clear, we first study, in this article this modified method for the advection equation. We point out the stability of this new method and the error estimate of the approximation solution is discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

hp–Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains

In this article, we develop the a posteriori error estimation of hp–version discontinuous Galerkin composite finite element methods for the discretization of second‐order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. Although standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. Computable bounds on the error measured in terms of a natural (mesh‐dependent) energy norm are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp–adaptive refinement procedure will be presented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1342–1367, 2014     

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.     

Weak solvability of interior transmission problems via mixed finite elements and Dirichlet-to-Neumann mappings

We study the weak solvability of an interior linear-nonlinear transmission problem arising in steady heat transfer and potential theory. For the variational formulation, we use a Dirichlet-to-Neumann mapping on the interface, which is obtained from the application of the boundary integral method to the linear domain, and we utilize a mixed finite element method in the nonlinear region. Existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete Galerkin method are derived. In particular, a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this Galerkin scheme.