This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two-grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B-spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order *α*, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

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, 2017

]]>In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In this article, we study the Drude models of Maxwell's equations in three-dimensional metamaterials. We derive new global energy-tracking identities for the three dimensional electromagnetic problems in the Drude metamaterials, which describe the invariance of global electromagnetic energy in variation forms. We propose the time second-order global energy-tracking splitting FDTD schemes for the Drude model in three dimensions. The significant feature is that the developed schemes are global energy-preserving, unconditionally stable, second-order accurate both in time and space, and computationally efficient. We rigorously prove that the new schemes satisfy these energy-tracking identities in the discrete form and the discrete variation form and are unconditionally stable. We prove that the schemes in metamaterials are second order both in time and space. The superconvergence of the schemes in the discrete *H*^{1} norm is further obtained to be second order both in time and space. Their approximations of divergence-free are also analyzed to have second-order accuracy both in time and space. Numerical experiments confirm our theoretical analysis results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

In this article, we extend the recently developed weak Galerkin method to solve the second-order hyperbolic wave equation. Many nice features of the weak Galerkin method have been demonstrated for elliptic, parabolic, and a few other model problems. This is the initial exploration of the weak Galerkin method for solving the wave equation. Here we successfully developed and established the stability and convergence analysis for the weak Galerkin method for solving the wave equation. Numerical experiments further support the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>We develop a general model describing a structured susceptible-infected (SI) population coupled with the environment. This model applies to problems arising in ecology, epidemiology, and cell biology. The model consists of a system of quasilinear hyperbolic partial differential equations coupled with a system of nonlinear ordinary differential equations that represents the environment. We develop a second-order high resolution finite difference scheme to numerically solve the model. Convergence of this scheme to a weak solution with bounded total variation is proved. Numerical simulations are provided to demonstrate the high-resolution property of the scheme and an application to a multi-host wildlife disease model is explored.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In this article, a fourth-order compact and conservative scheme is proposed for solving the nonlinear Klein-Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge-Kutta-Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete -norm. Numerical results show that the integral method with variational limit gives an efficient fourth-order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein-Gordon equation compared with other methods under the same condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>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, 2017

]]>This article studies a nonuniform finite difference method for solving the degenerate Kawarada quenching-combustion equation with a vibrant stochastic source. Arbitrary grids are introduced in both space and time via adaptive principals to accommodate the uncertainty and singularities involved. It is shown that, under proper constraints on mesh step sizes, the positivity, monotonicity of the solution, and numerical stability of the scheme developed are well preserved. Numerical experiments are given to illustrate our conclusions.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In this article, we obtain local energy and momentum conservation laws for the Klein-Gordon-Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy- and momentum-preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time-space region. With suitable boundary conditions, the schemes will be charge- and energy-/momentum-preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>We develop an a posteriori error estimator which focuses on the local *H*^{1} error on a region of interest. The estimator bounds a weighted Sobolev norm of the error and is efficient up to oscillation terms. The new idea is very simple and applies to a large class of problems. An adaptive method guided by this estimator is implemented and compared to other local estimators, showing an excellent performance.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

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. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three-dimensional space, with a view toward applications in plasma physics and fluid dynamics.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In this article, we consider a penalty finite element (FE) method for incompressible Navier-Stokes type variational inequality with nonlinear damping term. First, we establish penalty variational formulation and prove the well-posedness and convergence of this problem. Then we show the penalty FE scheme and derive some error estimates. Finally, we give some numerical results to verify the theoretical rate of convergence.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2017

]]>We present in this article a very adapted finite volume numerical scheme for transport type-equation. The scheme is an hybrid one combining an anti-dissipative method with down-winding approach for the flux (Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière, and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5 one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme able to capture in exact way the numerical solution of transport type-equation without artifact like numerical diffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution. This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behavior of the solution are of central importance in the modeling what is often the case in context of population dynamics where the final distribution of the considered population and its mass preservation relation are required for prediction.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>This work combines the consistency in lower-order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one-dimensional elliptic boundary value problem on nonuniform meshes, and a first-order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker-and-Cell scheme for the two-dimensional incompressible Stokes problem on unstructured meshes. A first-order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>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, 2017

]]>In this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non-oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton-Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

In this article, a kind of meshless local radial point interpolation (MLRPI) method is proposed to two-dimensional fractional-time convection-diffusion-reaction equations and satisfactory agreements are archived. This method is based on meshless methods and benefits from collocation ideas but it does not belong to the traditional global meshless collocation methods. In MLRPI method, it does not need any kind of integration locally or globally over small quadrature domains which is essential in the finite element method and those meshless methods based on Galerkin weak form. Also, it is not needed to determine shape parameter which plays important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of this kind of MLRPI method is less expensive. The stability and convergence of this meshless approach are discussed and theoretically proven. It is proved that the present meshless formulation is very effective for modeling and simulation of fractional differential equations. Furthermore, the numerical studies on sensitivity analysis and convergence analysis show the stability and reliable rates of convergence.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz–Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz–Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

]]>A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier–Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes a frequent requirement in the simulation of turbulent flows in complex geometries. The two-step method is considered here; to obtain a regularization that is second order accurate in space and time, the method computes a low-order accurate, stable, and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is stable and the accuracy in both space and time is improved after the correction step. We also perform a qualitative test to demonstrate that the method is capable of capturing qualitative features of a turbulent flow, even on a very coarse mesh.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>In this article, we deal with a rigorous error analysis for the finite element solutions of the two-dimensional Cahn–Hilliard equation with infinite time. The error estimates with respect to are proven for the fully discrete conforming piecewise linear element solution under Assumption (A1) on the initial value and Assumption (A2) on the discrete spectrum estimate in the finite element space. The analysis is based on sharp a-priori estimates for the solutions, particularly reflecting their behavior as . Numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the fully discrete mixed finite element methods.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an *H*^{1} technique are introduced to establish the optimal point-wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of
with mesh size *h* and time step *τ*. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in *L*^{2} and *H*^{1} norms. Results based on the usual finite element method are provided to confirm the theoretical estimates.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

In this article, numerical study for both nonlinear space-fractional Schrödinger equation and the coupled nonlinear space-fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz-Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>This survey enfolds rigorous analysis of the defect-correction finite element (FE) method for the time-dependent conduction-convection problem which based on the Crank-Nicolson scheme. The method consists of two steps: solve a nonlinear problem with an added artificial viscosity term on a FE grid and correct the solutions on the same grid using a linearized defect-correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with high Reynolds number is illustrated in several numerical experiments.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential, 2016

]]>In this article, we study a spectral meshless radial point interpolation of pseudoparabolic equations in two spatial dimensions. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high-order convergence rate. The time derivatives are approximated by the finite difference time-stepping method. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical results are presented to illustrate the theoretical findings.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>In this article, we propose and analyse a local projection stabilized and characteristic decoupled scheme for the fluid–fluid interaction problems. We use the method of characteristics type to avert the difficulties caused by the nonlinear term, and use the local projection stabilized method to control spurious oscillations in the velocities due to dominant convection, and use a geometric averaging idea to decouple the monolithic problems. The stability analysis is derived and numerical tests are performed to demonstrate the robustness of this new method.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential, 2016

]]>We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL-WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi-Lagrangian) scheme using a fixed velocity *v*. A flux correction computation accounts for particles that cross the *v*-traceline during the time step. If *v* = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when *v* is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps.

The *v*-traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re-averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

This is the final part of a series of articles where we have studied numerical instability (NI) of localized solutions of the generalized nonlinear Schrödinger equation (gNLS). It extends our earlier studies of this topic in two ways. First, it examines differences in the development of the NI between the case of the purely cubic NLS and the case where the gNLS has an external bounded potential. Second, it investigates how the NI is affected by the oscillatory dynamics of the simulated pulse. The latter situation is common when the initial condition is not an exact stationary soliton. We have found that in this case, the NI may remain weak when the time step exceeds the threshold quite significantly. This means that the corresponding numerical solution, while formally numerically unstable, can remain sufficiently accurate over long times, because the numerical noise will stay small.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>We prove long-time stability of linearly extrapolated BDF2 (BDF2LE) timestepping methods, together with finite element spatial discretizations, for incompressible Navier-Stokes equations (NSE) and related multiphysics problems. For the NSE, Boussinesq, and magnetohydrodynamics schemes, we prove unconditional long time *L*^{2} stability, provided external forces (and sources) are uniformly bounded in time. We also provide numerical experiments to compare stability of BDF2LE to linearly extrapolated Crank-Nicolson scheme for NSE, and find that BDF2LE has better stability properties, particularly for smaller viscosity values.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a posteriori estimators with a specifically tailored oscillation and show that, on two-dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decay rates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

]]>We analyze a multilevel diagonal additive Schwarz preconditioner for the adaptive coupling of FEM and BEM for a linear 2D Laplace transmission problem. We rigorously prove that the condition number of the preconditioned system stays uniformly bounded, independently of the refinement level and the local mesh-size of the underlying adaptively refined triangulations. Although the focus is on the nonsymmetric Johnson–Nédélec one-equation coupling, the principle ideas also apply to other formulations like the symmetric FEM-BEM coupling. Numerical experiments underline our theoretical findings. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015

]]>In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two-sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017

]]>Based on two-grid discretizations, a two-parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher-order finite element interpolants of the velocity into a lower-order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017

]]>Higher order non-Fickian diffusion theories involve fourth-order linear partial differential equations and their solutions. A quintic polynomial spline technique is used for the numerical solutions of fourth-order partial differential equations with Caputo time fractional derivative on a finite domain. These equations occur in many applications in real life problems such as modeling of plates and thin beams, strain gradient elasticity, and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical, and aerospace engineering. The quintic polynomial spline technique is used for space discretization and the time-stepping is done using a backward Euler method based on the *L*1 approximation to the Caputo derivative. The stability and convergence analysis are also discussed. The numerical results are given, which demonstrate the effectiveness and accuracy of the numerical method. The numerical results obtained in this article are also compared favorably well with the results of (S. S. Siddiqi and S. Arshed, Int. J. Comput. Math. 92 (2015), 1496–1518). © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 445–466, 2017

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

Strong convergence of the numerical solution to a weak solution is proved for a nonlinear coupled flow and transport problem arising in porous media. The method combines a mixed finite element method for the pressure and velocity with an interior penalty discontinuous Galerkin method in space for the concentration. Using functional tools specific to broken Sobolev spaces, the convergence of the broken gradient of the numerical concentration to the weak solution is obtained in the *L*^{2} norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 489–513, 2017

We present a Waveform Relaxation (WR) version of the Neumann–Neumann algorithm for the wave equation in space-time. The method is based on a nonoverlapping spatial domain decomposition, and the iteration involves subdomain solves in space-time with corresponding interface conditions, followed by a correction step. Using a Fourier-Laplace transform argument, for a particular relaxation parameter, we prove convergence of the algorithm in a finite number of steps for the finite time intervals. The number of steps depends on the size of the subdomains and the time window length on which the algorithm is employed. We illustrate the performance of the algorithm with numerical results, followed by a comparison with classical and optimized Schwarz WR methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 514–530, 2017

]]>This article deals with the analytic and numerical stability of numerical methods for a parabolic partial differential equation with piecewise continuous arguments of alternately retarded and advanced type. First, application of the theory of separation of variables in matrix form and the Fourier method, the necessary and sufficient condition under which the analytic solution is asymptotically stable is derived. Then, the *θ*-methods are applied to solve the corresponding initial value problem, the sufficient conditions for the asymptotic stability of numerical methods are obtained. Finally, several numerical examples are presented to support the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 531–545, 2017

Two-grid variational multiscale (VMS) algorithms for the incompressible Navier-Stokes equations with friction boundary conditions are presented in this article. First, one-grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two-grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017

]]>We derive residual-based *a posteriori* error estimates of finite element method for linear parabolic interface problems in a two-dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving *a posteriori* estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish *a posteriori* error estimates with optimal order convergence in the -norm and almost optimal order in the -norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017