An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

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     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

A posteriori error estimation of the p‐Laplace problem

Two a posteriori error estimates are discussed for the p‐Laplace problem. Up to errors in their numerical computation, they provide a guaranteed upper bound for the W^1,p‐seminorm and a weighted W^1,2‐seminorm of u – u_h. The first, sharper a posteriori estimate is based on the numerical solution of local interface problems. The standard residual‐based error estimate is addressed with emphasis on involved constants, determined as local eigenvalues. Numerical examples that illustrate the performance of these estimators can be found in [3].     

Robust partially mode‐dependent H∞ filtering for discrete‐time nonhomogeneous Markovian jump neural networks with additive gain perturbations

This paper studies the robust partially mode‐dependent H_∞ filtering for nonhomogeneous Markovian jump neural networks with additive gain perturbations. The discrete time‐varying jump transition probability matrix is considered to be a polytope set. A partially mode‐dependent filter with additive gain perturbations is constructed to increase the robustness of the filter, which is subjects to H_∞ performance index. Based on the Lyapunov function approach, sufficient conditions are established such that the filtering error system is robustly stochastically stable. The efficiency of the new technique is illustrated by an illustrative example and a biological network example.     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

Nitsche type mortaring for elliptic problems with corner singularities

The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non‐matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet conditions for the case that the interface passes re‐entrant corners of the domain and local mesh refinement is applied. Some properties of the finite element scheme and error estimates in a discrete H¹‐like and in the L₂‐norm are proved.     

Operator splitting for numerical solution of the modified Burgers' equation using finite element method

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L₂ and L_∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method.     

Uniform superconvergence of a Galerkin finite element method on Shishkin‐type meshes

We consider a Galerkin finite element method that uses piecewise bilinears on a class of Shishkin‐type meshes for a model singularly perturbed convection‐diffusion problem on the unit square. The method is shown to be convergent, uniformly in the diffusion parameter ϵ, of almost second order in a discrete weighted energy norm. As a corollary, we derive global L₂‐norm error estimates and local L_∞‐norm estimates. Numerical experiments support our theoretical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16:426–440, 2000     

Second‐order finite‐volume schemes for a non‐linear hyperbolic equation: error estimate

We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation u_t(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u₀. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u₀ ∈ L^∞ ∩ BV_loc(ℝ^N), we get an error estimate of order h^1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

A (T,ψ)‐ψe decoupled scheme for a time‐dependent multiply‐connected eddy current problem

The aim of this paper is to develop a fully discrete ( T ,ψ)‐ψ_e finite element decoupled scheme to solve time‐dependent eddy current problems with multiply‐connected conductors. By making ‘cuts’ and setting jumps of ψ_e across the cuts in nonconductive domain, the uniqueness of ψ_e is guaranteed. Distinguished from the traditional T ‐ ψ method, our decoupled scheme solves the potentials T and ψ‐ ψ_e separately in two different simple equation systems, which avoids solving a saddle‐point equation system and leads to a remarkable reduction in computational efforts. The energy‐norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley & Sons, Ltd.     

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h⁴ + Δt²) . Stability of the technique is examined using von‐Neumann analysis. L₂ and L_∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time.     

An expanded mixed covolume method for sobolev equation with convection term on triangular grids

The expanded mixed covolume method for the two‐dimensional Sobolev equation with convection term is developed and studied. This method uses the lowest‐order Raviart‐Thomas mixed finite element space as the trial function space. By introducing a transfer operator γ_h which maps the trial function space into the test function space and combining expanded mixed finite element with mixed covolume method, the continuous‐in‐time, discrete‐in‐time expanded mixed covolume schemes are constructed, and optimal error estimates for these schemes are obtained. Numerical results are given to examine the validity and effectiveness of the proposed schemes.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

Adini‐Q1‐P3 FEM for general elastic multi‐structure problems

An Adini‐Q₁‐P₃ finite element method is introduced to solve general elastic multi‐structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, and transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1092–1112, 2011     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c₀ is strictly positive and when c₀ is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013