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     

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     

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     

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     

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     

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     

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     

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     

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     

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     

Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and optimal estimates of are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017     

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     

Decoupled schemes for unsteady MHD equations. I. time discretization

In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional stability and convergence of this scheme are provided. The optimal error estimates for velocity and magnet are provided, and the optimal error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017     

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     

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

Circular symmetrization,subordination and arclength problems on convex functions

We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .     

Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier–Stokes system

We consider a mixed finite‐volume finite‐element method applied to the Navier–Stokes system of equations describing the motion of a compressible, barotropic, viscous fluid. We show convergence as well as error estimates for the family of numerical solutions on condition that: (a) the underlying physical domain as well as the data are smooth; (b) the time step and the parameter of the spatial discretization are proportional, ; and (c) the family of numerical densities remains bounded for . No a priori smoothness is required for the limit (exact) solution. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1208–1223, 2017     

A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids

In this article, we develop and analyze a new recovery‐based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the ‐norm of the solution is of order , when tensor product polynomials of degree at most are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be . We use our derivative recovery result to develop a robust recovery‐type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery‐type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery‐type a posteriori error estimates, at a fixed time, converge to the true errors in the ‐norm under mesh refinement. The order of convergence is proved to be . Our theoretical results are valid for piecewise polynomials of degree and under the condition that each component, , of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery‐based a posteriori error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1224–1265, 2017     

Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus

We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation with peaks on a Clifford type torus. Here where with for all Each is a function and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a or a process.