An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

An efficient numerical method for the resolution of the Kirchhoff‐Love dynamic plate equation

We solve numerically the Kirchhoff‐Love dynamic plate equation for an anisotropic heterogeneous material using a spectral method. A mixed velocity‐moment formulation is proposed for the space approximation allowing the use of classical Lagrange finite elements. The benefit of using high order elements is shown through a numerical dispersion analysis. The system resulting from this spatial discretization is solved analytically. Hence this method is particularly efficient for long duration experiments. This time evolution method is compared with explicit and implicit finite differences schemes in terms of accuracy and computation time. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Fast solvers with block‐diagonal preconditioners for linear FEM–BEM coupling

The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block‐diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

This study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method.     

Applications of cubic B‐splines collocation method for solving nonlinear inverse parabolic partial differential equations

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B‐splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B‐splines for spatial variable and derivatives, which produce an ill‐posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 88–104, 2017     

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

In this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017     

A FE‐BE coupling for a fluid‐structure interaction problem: Hierarchical a posteriori error estimates

In this article, we establish a hierarchical a posteriori error estimate for a coupling of finite elements and boundary elements for a fluid‐structure interaction problem posed in two and three dimensions. These methods combine boundary elements for the exterior fluid and finite elements for the elastic structure. We consider two weak formulations, a nonsymmetric one and a symmetric one, which are both uniquely solvable. We present the reliability and efficiency of the error estimates. For the two dimensional case, we compute local error indicators which allow us to develop an adaptive mesh refinement strategy on triangles. For the three dimensional case, we use hexahedrons as elements. Numerical experiments underline our theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

An extension of the Gegenbauer pseudospectral method for the time fractional Fokker‐Planck equation

The time fractional Fokker‐Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial‐boundary value problems of the time fractional Fokker‐Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.     

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.     

Wavelet-Galerkin method for solving parabolic equations in finite domains

A novel wavelet-Galerkin method tailored to solve parabolic equations in finite domains is presented. The emphasis of the paper is on the development of the discretization formulations that are specific to finite domain parabolic equations with arbitrary boundary conditions based on weak form functionals. The proposed method also deals with the development of algorithms for computing the associated connection coefficients at arbitrary points. Here the Lagrange multiplier method is used to enforce the essential boundary conditions. The numerical results on a two-dimensional transient heat conducting problem are used to validate the proposed wavelet-Galerkin algorithm as an effective numerical method to solve finite domain parabolic equations.     

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     

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A Locally Conservative Eulerian-Lagrangian Finite Difference Method for a Parabolic Equation

The object of this paper is to define a finite difference analogue of a locally conservative Eulerian—Lagrangian method based on mixed finite elements and to prove its convergence. The method is appropriate for convection-dominated diffusive processes; here, it will be considered in the case of a semilinear parabolic equation in a single space variable.This revised version was published online in October 2005 with corrections to the Cover Date.     

A fourth‐order compact algorithm for nonlinear reaction‐diffusion equations with Neumann boundary conditions

In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction‐diffusion equations is fourth‐order accurate in both the temporal and spatial dimensions. We also prove that the standard second‐order approximation to zero Neumann boundary conditions provides fourth‐order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.