Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Rapid Solution of Minimal Riesz Energy Problems

In , , we compute the solution to both the unconstrained and constrained Gauss variational problem, considered for the Riesz kernel of order and a pair of compact, disjoint, boundaryless ‐dimensional ‐manifolds , , where , each being charged with Borel measures with the sign prescribed. Such variational problems over a cone of Borel measures can be formulated as minimization problems over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space , where and (see Harbrecht et al., Math. Nachr. 287 (2014), 48–69). We thus approximate the sought density by piecewise constant boundary elements and apply the primal‐dual active set strategy to impose the desired inequality constraints. The boundary integral operator which is defined by the Riesz kernel under consideration is efficiently approximated by means of an ‐matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first‐order optimality system. Numerical results in are given to demonstrate our approach. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1535–1552, 2016     

Two unconditionally stable and convergent difference schemes with the extrapolation method for the one‐dimensional distributed‐order differential equations

The Grünwald formula is used to solve the one‐dimensional distributed‐order differential equations. Two difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders and in maximum norm, respectively, where and are step sizes in time, space and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders and respectively. An illustrative numerical example is given to confirm the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 591–615, 2016     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Electromagnetic eigenfields in checkerboard patterns

Here are described four solvers for time‐harmonic electromagnetic fields in checkerboard patterns. A pattern is built by four squares with constant permittivity, or . It is enclosed by conducting walls or is a unit cell of a periodic structure. The field is represented in two ways: by , the transverse component of the magnetic induction, and by , the magnetic vector potential in Lorenz gauge. and satisfy Helmholtz equations in each square as well as transmission and boundary conditions (BCs). These governing equations yield eigensolutions and , which are found to be at worst. Variational versions of the governing equations are introduced. The weak formulations for are standard, while those for are new. They imply that the derivative transmission and BCs are satisfied weakly on interfaces between regions with different permittivity. Eigenpairs are computed approximately by spectral element methods. They yield mutually consistent eigenpairs. However, only about half of the eigenpairs () correspond to eigenpairs (). For each set of BCs, the first few eigenfrequencies are given by tables, and some of the eigenfunctions are presented by contour plots. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 418–444, 2016     

The local superconvergence of the linear finite element method for the poisson problem

Assume that . In this study, the Richardson extrapolation for the tensor‐product block element and the linear finite element theory of the Green's function will be combined to study the local superconvergence of finite element methods for the Poisson equation in a bounded polytopic domain (polygonal or polyhedral domain for ), where a family of tensor‐product block partitions is not required or the solution need not have high global smoothness. We present a special family of partitions satisfying, for any , e is a tensor‐product block whenever where denotes the distance between e and . By the linear finite element theory of the Green's function and the Richardson extrapolation for the tensor‐product block element, we obtain the local superconvergence of the displacement for the linear finite element method over the special family of partitions . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 930–946, 2014     

On a nonoverlapping additive Schwarz method for h‐p discontinuous Galerkin discretization of elliptic problems

The condition number of a discontinuous Galerkin finite element discretization preconditioned with a nonoverlapping additive Schwarz method is analyzed. We improve the result of Antonietti and Houston (J Sci Comput 46 (2011), 124–149), where a bound has been proved for a two‐level nonoverlapping additive Schwarz method with coarse problem using polynomials of degree on a coarse mesh size . In a more general framework, where the concurrency of the algorithm is increased by applying solvers on subdomains smaller than the coarse grid cells, we prove that the condition number of the preconditioned system is where is the coarse space element degree polynomial and is the size of subdomain where local problems are solved in parallel. Our result also extends to the case of discontinuous coefficient, piecewise constant on the coarse grid, for a composite continuous–discontinuous Galerkin discretization. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1572–1590, 2016     

Uniform stability of spectral nonlinear Galerkin methods

This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, u₀, f). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δt ≤ (2/λλ₁). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δt = O(λ) and Δt = O(λ), respectively, where λ ≤ λ, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Finite element approximation of a phase field model arising in nanostructure patterning

We consider a fully practical finite element approximation of the nonlinear parabolic Cahn–Hilliard system subject to an initial condition on the conserved order parameter , and mixed boundary conditions. Here, is the interfacial parameter, is the field strength parameter, is the obstacle potential, is the diffusion coefficient, and denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential and is the electrostatic potential. The system, in the context of nanostructure patterning, has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field. In the limit , it reduces to a sharp interface problem that models the evolution of an unstable interface between two dielectric media in the presence of a quasistatic electric field. On introducing a finite element approximation for the above Cahn–Hilliard system, we prove existence and stability of a discrete solution. Moreover, in the case of two space dimensions, we are able to prove convergence and hence existence of a solution to the considered system of partial differential equations. We demonstrate the practicality of our finite element approximation with several numerical simulations in two and three space dimensions. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1890–1924, 2015     

The time discretization in classes of integro‐differential equations with completely monotonic kernels: Weighted asymptotic convergence

We consider the time discretization for the solution of the equation with . Here, the operators L_j are densely defined positive self‐adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity in H . The kernel functions , are assumed to be completely monotonic on (0,∞) and locally integrable, but not constant. The considered time discretization method comes from [Da Xu, Science China Mathematics 56 (2013), 395–424], where the backward Euler method is combined with order one convolution quadrature for approximating the integral term. In this article, the convergence properties of the time discretization are given in the weighted and norm, where ρ is a given weighted function. Numerical experiments show the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 896–935, 2016     

Superconvergence for a time‐discretization procedure for the mixed finite element approximation of miscible displacement in porous media

A time‐stepping procedure is established and analyzed for the problem of miscible displacement in porous medium. The fluid velocity based on mixed element method is post‐processed by interpolation along Gauss lines. Error analysis shows that this procedure has a superconvergence error bound O(h) with respect to the parameter h_p, which is one order higher than the conventional Galerkin or mixed finite element procedures. Compared with the results in the references, the parameter constraint conditions in this article are obviously relaxed. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Least‐squares spectral method for velocity‐vorticity‐pressure form of the Stokes equations

The aim of this article is to present and analyze first‐order system least‐squares spectral method for the Stokes equations in two‐dimensional spaces. The Stokes equations are transformed into a first‐order system of equations by introducing vorticity as a new variable. The least‐squares functional is then defined by summing up the ‐ and ‐norms of the residual equations. The ‐norm in the least‐squares functional is replaced by suitable operator. Continuous and discrete homogeneous least‐squares functionals are shown to be equivalent to ‐norm of velocity and ‐norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 661–680, 2016     

Group representations that resist random sampling

We show that there exists a family of groups G_n and nontrivial irreducible representations ρ_n such that, for any constant t, the average of ρ_n over t uniformly random elements has operator norm 1 with probability approaching 1 as . More quantitatively, we show that there exist families of finite groups for which random elements are required to bound the norm of a typical representation below 1. This settles a conjecture of A. Wigderson. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 605–614, 2015     

Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q₂ ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q₃ ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q₂‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q₂ ? P discretization, and the Q₃ ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

We present a fourth‐order Hermitian box‐scheme (HB‐scheme) for the Poisson problem in a cube. A single‐nonstaggered regular grid is used supporting the discrete unknowns u and . The scheme is fourth‐order accurate for u and in norm. The fast numerical resolution uses a matrix capacitance method, resulting in a computational complexity of . Numerical results are reported on several examples including nonseparable problems. The present scheme is the extension to the three‐dimensional case of the HB‐scheme presented in Abbas and Croisille [J Sci Comp 49 (2011), 239–267]. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 609–629, 2015     

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation

In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L² error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L²‐norm under mesh refinement. The order of convergence is proved to be , when p‐degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L²‐norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015     

Superconvergence of interpolated collocation solutions for Hammerstein equations

In this article, we discuss the superconvergence of the interpolated collocation solutions for Hammerstein equations. Applying this new interpolation postprocessing to the collocation approximation x_h, we get a higher accuracy approximation I x_h, whose convergence order is the same as that of the iterated collocation method. Such an interpolation postprocessing method is much simpler. Also, numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Numerical solution of one‐dimensional Burgers' equation

In this article, an iterative method for the approximate solution of a class of Burgers' equation is obtained in reproducing kernel space . It is proved the approximation converges uniformly to the exact solution u(x, t) for any initial function under trivial conditions, the derivatives of are also convergent to the derivatives of u(x, t), and the approximate solution is the best approximation under the system © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1251–1264, 2015