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Preview: Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Wiley Online Library : Numerical Methods for Partial Differential Equations

Published: 2017-11-01T00:00:00-05:00


The fictitious domain method with L2-penalty for the Stokes problem with the Dirichlet boundary condition


We consider the fictitious domain method with L2-penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order O ( ϵ 1 4 ) in H1-norm for the velocity and in L2-norm for the pressure, where ϵ is the penalty parameter. The L2-norm error estimate for the velocity is upgraded to O ( ϵ ) . Moreover, we derive the a priori estimates depending on ϵ for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate O ( h + ϵ 1 4 ) in H1-norm for the velocity and in L2-norm for the pressure, where h denotes the discretization parameter. For the velocity in L2-norm, the convergence rate is improved to O ( h + ϵ 1 2 ) . The theoretical results are verified by the numerical experiments.

Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer


We consider a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion. We propose and analyze a finite volume scheme based on two-point flux approximation with upwind mobilities. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. Based on these nonlinear stability results, we show that the scheme converges toward a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.

High-order Padé and singly diagonally Runge-Kutta schemes for linear ODEs, application to wave propagation problems


In this article, we address the problem of constructing high-order implicit time schemes for wave equations. We consider two classes of one-step A-stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge-Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge-Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second-class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high-order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.

Heat and mass transfer in unsteady MHD slip flow of Casson fluid over a moving wedge embedded in a porous medium in the presence of chemical reaction: Numerical Solutions using Keller-Box Method


The hydromagnetic mixed convection flow of Casson fluid due to moving wedge embedded in a porous medium in the presence of chemical reaction and viscous dissipation is investigated. The Joule heating due to a magnetic field and porous medium heating is also considered. Similarity transformations are utilized to convert nonlinear partial differential equations into nonlinear ordinary differential equations. The resulting equations are solved numerically via Keller-box method. The numerical results are achieved for limiting cases and are revealed in close agreement with those of the results available in the literature. It is noticed that unsteadiness parameter thinning the velocity boundary layer while opposite to this was found in the thermal and concentration boundary layers. The dimensionless temperature is observed to be enhanced with increment in Eckert number.

Superconvergence analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation


The purpose of this article is to apply E Q 1 rot nonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of E Q 1 rot nonconforming FE (see (2.3) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank-Nicolson fully discrete scheme is constructed and the superclose error estimate of order O ( h 2 + τ 2 ) for original variable u in broken H1-norm is also derived by using the properties of E Q 1 rot element and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

On solutions to the second-order partial differential equations by two accurate methods


In this article, we investigate the reproducing kernel method and the difference schemes method for solving the second-order partial differential equations. Numerical results have been shown to prove the efficiency of the methods. Results prove that the methods are very effective.

Efficient hybrid method for solving special type of nonlinear partial differential equations


In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two- and three-dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.

A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation


An efficient spectral method is developed in this paper for the two-dimensional Stokes eigenvalues on arbitrary triangle. By using the spectral theory of compact operator and approximate property of orthogonal polynomial, we give the error estimate of the approximate eigenvalues and eigenfunctions. In addition, we also present some numerical results to show the validity of our algorithm and the correctness of the theoretical results.

Superconvergence of Ritz-Galerkin finite element approximations for second order elliptic problems


In this paper, the author derives an O ( h 4 ) -superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second-order elliptic equation − ∇ · ( A ∇ u ) = f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A -equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation


In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1-type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data


In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier-Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier-Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier-Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Numerical simulation of time fractional Cable equations and convergence analysis


In this article, the numerical solution of time fractional Cable equation is considered. We convert the time fractional Cable equations into equivalent integral equations with singular kernel, then propose a spectral collection method in both time and space discretizations with a spectral expansion of Lagrange interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results.

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers


The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 106. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.

Dynamical study of two predators and one prey system with fractional Fourier transform method


In this work, we investigate both the analytical and numerical studies of the dynamical model comprising of three species systems. We analyze the linear stability of stationary solutions in the one-dimensional multisystem modeling the interactions of two predators and one prey species. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. The analysis results presented have established the possibility of the three-interacting species to coexist harmoniously, this feat is achieved by combining the local and global analyses to determine the global dynamics of the system. In the presence of a fractional diffusion term, we introduced a fractional Fourier transform for solving the system modeled by fractional partial differential equations. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator with exponential accuracy and a completely straightforward extension to high spatial dimensions. The scheme is described in detail and justified by a number of computational experiments.

A two-grid stabilized mixed finite element method for Darcy-Forchheimer model


A two-grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two-dimensional Darcy-Forchheimer model. We use the derivative of a smooth function, ε 2 + u 2 , to approximate the derivative of | u | in constructing the two-grid algorithm. The two-grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two-grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.

Combining MFS and PGD methods to solve transient heat equation


We propose in this article a numerical algorithm based on the combination of the method of fundamental solutions (MFS) and the proper generalized decomposition technique (PGD) to solve time-dependent heat equation. The MFS is considered as a truly meshless technique well adapted for a wide range of physical problems and the PGD approach can be considered as a reduction technique based on the separated representation of the variable functions. The proposed study relates to a separation between the spatial and temporal coordinates. To show the effectiveness of the proposed algorithm, several examples are presented and compared to the reference results.

Application of finite difference method of lines on the heat equation


In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three-dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.

The method of mixed volume element-characteristic mixed volume element and its numerical analysis for three-dimensional slightly compressible two-phase displacement


Numerical simulation of oil-water two-phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection-diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in l 2 norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems.

New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves


In this article, some high-order accurate difference schemes of dispersive shallow water waves with Rosenau-KdV-RLW-equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is O ( h 4 + k 2 ) in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

Maximum norm stability and error estimates for the evolving surface finite element method


We show convergence in the natural L ∞ and W 1 , ∞ norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this, we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.

New two step Laplace Adam-Bashforth method for integer a noninteger order partial differential equations


This article presents a novel method that allows to generalize the use of the Adam-Bashforth to Partial Differential Equations with local and nonlocal operator. The Method derives a two step Adam-Bashforth numerical scheme in Laplace space and the solution is taken back into the real space via inverse Laplace transform. The method yields a powerful numerical algorithm for fractional order derivative where the usually very difficult to manage summation in the numerical scheme disappears. Error Analysis of the method is also presented. Applications of the method and numerical simulations are presented on a wave–equation-like, and on a fractional order diffusion equation.

Numerical solution for high-order linear complex differential equations with variable coefficients


In this paper, we have obtained the numerical solutions of complex differential equations with variable coefficients by using the Legendre Polynomials and we have performed it on two test problems. Then, we applied with different technical of error analysis to the test problems. When we compared exact solutions and numerical solutions of tables and graphs, we realized that our method is reliable, practical, and functional.

Two-grid method for two-dimensional nonlinear Schrödinger equation by finite element method


A conservative two-grid finite element scheme is presented for the two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse-grid solution as the initial guess. Moreover, error estimates are conducted for the two-grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 2 ) in the two-grid method. The numerical results show that this method is very effective.

Chaos in a nonlinear Bloch system with Atangana–Baleanu fractional derivatives


In this article, a nonlinear model of the Bloch equation to include both fractional derivatives with variable-order, constant-order, and time delays was considered. The fractional derivative with the generalized Mittag-Leffler function as kernel is introduced due to the nonlocality of the dynamical system. To find a numerical solution of the delay variable-order model, a predictor corrector method had been developed to solve this system. The existence and uniqueness of the numerical scheme was discussed in detail. For the constant-order, we presented the existence and uniqueness of a positive set of the solutions for the new model and the Adams–Moulton rule was considered to solved numerically the fractional equations. The behavior of the fractional commensurate order nonlinear delay-dependent Bloch system with total order less than 3, which exhibits chaos and transient chaos, was presented. In addition, it is found that the presence of fractional variable-order in the nonlinear Bloch system exhibit more complicated dynamics can improve the stability of the solutions.

High-order implicit staggered-grid finite differences methods for the acoustic wave equation


Motivated by the idea that staggered-grid methods give a greater stability and give energy conservation, this article presents a new family of high-order implicit staggered-grid finite difference methods with any order of accuracy to approximate partial differential equations involving second-order derivatives. In particular, we numerically analyze our new methods for the solution of the one-dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ( 2 M + 2 ) -point operator: the explicit formula is ( 2 M ) th-order and the implicit formula is ( 2 M + 2 ) th-order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.

Orthogonal spline collocation scheme for multiterm fractional convection-diffusion equation with variable coefficients


The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two-dimensional multiterm fractional convection-diffusion reaction equation with variable coefficients. The optimal estimates in Hj (j = 0, 1, 2) norms at each time step are derived. Also, L ∞ estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.

Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions


Recently, many new applications in engineering and science are governed by a series of time-fractional partial integrodifferential equations, which will lead to new challenges for numerical simulation. In this article, we propose and analyze an efficient iterative algorithm for the numerical solutions of such equations subject to initial and Dirichlet boundary conditions. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n -term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such integrodifferential equations.

An Investigation on Reliable Analytical and Numerical Methods for the Riesz Fractional Nonlinear Schrödinger Equation inQuantum Mechanics


In the article, a nonlinear Schrödinger equation with the Riesz fractional derivative has been considered. This equation has been solved by two reliable methods to investigate the accuracy of the solutions. In the implicit finite difference numerical scheme, the fractional centered difference is utilized to approximate the Riesz fractional derivative. Also, a novel modified optimal homotopy asymptotic method with Fourier transform (MOHAM-FT) has been proposed to compute the approximate solution of Riesz fractional nonlinear Schrödinger equation(RFNLSE). Further the numerical solutions of RFNLSE obtained by proposed implicit finite difference method, have been compared with that obtained by MOHAM-FT to exhibit the effectiveness of the suggested methods. Finally, the obtained solutions have been presented graphically to justify the efficiency of the methods.

Solving the burgers' and regularized long wave equations using the new perturbation iteration technique


In this study, an efficient framework is provided to handle nonlinear partial differential equations by implementing perturbation iteration method. This method is recovered and amended to solve the Burgers' and regularized long wave equations. Comparing our new solutions with the exact solutions reveals that this technique is extremely accurate and effective in solving nonlinear models. Convergence analysis and error estimate are also supplied using some critical theorems.

Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-Baleanu fractional model


The impact of magnetic field on Engine Oil based generalized Brinkman-type nanofluid over an oscillating vertical plate embedded in a porous medium is studied. Molybdenum Disulphide (MoS2) nanoparticles of spherical shape are suspended in Engine Oil, taken as conventional base fluid. Effect of thermal radiation in energy equation is also considered. A generalized model of Brinkman-type fluid is considered with newly introduced fractional derivatives known as Atangana-Baleanu Derivative (ABD) in the presence of heat transfer due to convection. Exact solution of the problem is determined by means of the Laplace transform. Expressions for velocity and temperature are obtained in terms of Mittag-Leffler and General Wright function. The effects of various pertinent parameters on velocity are portrayed and discussed graphically. Numerical results of rate of heat transfer are computed in tabular form. Which showed that increasing values of volume fraction and Prandtl number increase rate of heat transfer.

Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative


A one-dimensional steady-state advection-diffusion problem using summation-by-parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes are equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments.

Higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations


In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential-difference convection-diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time-dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of O ( Δ t + N − 2 ( ln ⁡ N ) 2 ) , where Δ t and N denote the time step and number of mesh-intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of O ( Δ t 2 + N − 2 ( ln ⁡ N ) 2 ) . Some numerical tests are performed to illustrate the high-order accuracy and parameter uniform convergence obtained with the proposed numerical methods.

The Hunter-Saxton Equation: A Numerical Approach Using Collocation Method


In this study, we are going to present an overview on the Hunter-Saxton equation which is a famous equation modelling waves in a massive director field of a nematic liquid crystal. The collocation finite element method is based on quintic B-spline basis for obtaining numerical solutions of the equation. Using this method, after discretization, solution of the equation expressed as linear combination of shape functions and B-spline basis. So, Hunter-Saxton equation converted to nonlinear ordinary differential equation system. With the aid of the error norms L 2 and L ∞ , some comparisons are presented between numeric and exact solutions for different step sizes. As a result, the authors observed that the method is a powerful, suitable and reliable numerical method for solving various kind of partial differential equations.

Local error estimates of the finite element method for an elliptic problem with a Dirac source term


The solutions of elliptic problems with a Dirac measure right-hand side are not H 1 in dimension d ∈ { 2 , 3 } and therefore the convergence of the finite element solutions is suboptimal in the L 2-norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the H s-norm, for s ≥ 1 on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in H 1-norm is shown on a family of quasiuniform meshes. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.

A posteriori error analysis for solving the Navier-Stokes problem and convection-diffusion equation


In this article, we consider the finite element discretization of the Navier-Stokes problem coupled with convection-diffusion equations where both the viscosity and the diffusion coefficients depend on the temperature. Existence and uniqueness of a solution are established. We prove a posteriori error estimates.

Superconvergent estimates of conforming finite element method for nonlinear time-dependent Joule heating equations


In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H1-norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis.

Long-time stability and asymptotic analysis of the IFE method for the multilayer porous wall model


In this article, we study the long-time stability and asymptotic behavior of the immersed finite element (IFE) method for the multilayer porous wall model for the drug-eluting stents. First, with the IFE method for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multilayer porous wall model converges to the corresponding elliptic equation if f ( x , t ) approaches to a steady-state f ¯ ( x ) in both L 1 ( 0 , t ; L 2 ( Ω ) ) and L ∞ ( 0 , t ; L 2 ( Ω ) ) norms as t + ∞ . Finally, some numerical experiments are given to verify the theoretical predictions.

A scattering-based algorithm for wave propagation in one dimension


We present an explicit numerical scheme to solve the variable coefficient wave equation in one space dimension with minimal restrictions on the coefficient and initial data.

Supercloseness analysis and polynomial preserving Recovery for a class of weak Galerkin Methods


In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results.

Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative


This article deals with the design, analysis, and implementation of a robust numerical scheme when applied to time-fractional reaction-diffusion system. Stability analysis and numerical treatment of chaotic fractional differential system in Riemann-Liouville sense are considered in this article. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise. Numerical results presented agree with the theoretical findings.

Numerical simulation and solutions of the two-component second order KdV evolutionarysystem


In this study, with the aid of Wolfram Mathematica 11, the modified exp ( − Ω ( η ) ) -expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ( − Ω ( η ) ) -expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.

A stabilized finite element method for a fictitious domain problem allowing small inclusions


The purpose of this work is to approximate numerically an elliptic partial differential equation posed on domains with small perforations (or inclusions). The approach is based on the fictitious domain method, and as the method's interest lies in the case in which the geometrical features are not resolved by the mesh, we propose a stabilized finite element method. The stabilization term is a simple, non-consistent penalization that can be linked to the Barbosa-Hughes approach. Stability and convergence are proved, and numerical results confirm the theory.

A block-centered finite difference method for fractional Cattaneo equation


In this article, a block-centered finite difference method for fractional Cattaneo equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete L 2 norm with optimal order of convergence O ( Δ t 3 − α + h 2 + k 2 ) both for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids


In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long-time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality-type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems


In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H2 norm error estimate between the multiscale solution and the homogenized solution with the first-order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log-normal coefficients. Their results demonstrate the efficiency of the proposed method.

Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu


In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. It was concluded by some eminent researchers that the Riemann-Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann-Liouville definition, from power-law kernel to generalized Mittag-Leffler-law via exponential-decay-law.

Unconditional superconvergence analysis of an H1-galerkin mixed finite element method for nonlinear Sobolev equations


An efficient H1-Galerkin mixed finite element method (MFEM) is presented with E Q 1 rot and zero order Raviart-Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order O ( h 2 ) for the original variable u in a broken H1 norm and the auxiliary variable q = a ( u ) ∇ u t + b ( u ) ∇ u in H ( div ; Ω ) norm are deduced without the boundedness of the numerical solution in L ∞ -norm. Conversely, a linearized Crank-Nicolson fully discrete scheme with the unconditional super close property O ( h 2 + τ 2 ) is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection-diffusion equations


This paper is devoted to developing an Il'in-Allen-Southwell (IAS) parameter-uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection-diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second-order accurate in the sense that it is second-order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first-order convergent in the discrete maximum norm uniformly in ϵ.

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials


In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Existence of solution of a finite volume scheme preserving maximum principle for diffusion equations


In this article, a cell-centered finite volume scheme preserving maximum principle for diffusion equations with scalar coefficients is developed. The construction of the scheme consists of three steps: at first the discrete normal flux is obtained by a linear combination of two single-sided fluxes, then the tangential term of the normal flux is modified by using a nonlinear combination of two single-sided tangential fluxes, finally the auxiliary unknowns in the tangential fluxes are calculated by the convex combinations of the cell-centered unknowns. It is proved that this nonlinear scheme satisfies the discrete maximum principle (DMP). Moreover, the existence of a solution of the nonlinear scheme is proved by using the Brouwer's fixed point theorem and the bounded estimates. Numerical experiments are presented to show that the scheme not only satisfies DMP, but also obtains the second-order accuracy and conservation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Stability analysis for fractional-order partial differential equations by means of space spectral time Adams-Bashforth Moulton method


In this article, a new numerical scheme space Spectral time Fractional Adam Bashforth Moulton method for the solution of fractional partial differential equations is offered. The proposed method is obtained by modifying, in a suitable way; the spectral technique and the method of lines. The attention is focused on the stability properties and hence an elegant stability analysis for the current approach is also provided. Finally, two examples are presented to illustrate the effectiveness of the reported method. Obtained results confirm the convergence and spectral accuracy of the proposed method in both space and time. In addition, a comparison with the existing studies is also made as a limiting case of the considered problem at the end and found in good agreement.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Numerical approximation using evolution PDE variational splines


This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

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Stable numerical solution of the Cauchy problem for the Laplace equation in irregular annular regions


This article is mainly concerned with the numerical study of the Cauchy problem for the Laplace equation in a bounded annular region. To solve this ill-posed problem, we follow a variational approach based on its reformulation as a boundary control problem, for which the cost function incorporates a penalized term with the input data. The cost function is minimized by a conjugate gradient method in combination with a finite element discretization. In the case where the input data is noisy, some preliminary error estimates, show that the penalization parameter may be chosen like the inverse of the level of noise. Numerical solutions in simple and complex domains show that this methodology produces stable and accurate solutions.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1799–1822, 2017

A fast numerical method for solving coupled Burgers' equations


A new fast numerical scheme is proposed for solving time-dependent coupled Burgers' equations. The idea of operator splitting is used to decompose the original problem into nonlinear pure convection subproblems and diffusion subproblems at each time step. Using Taylor's expansion, the nonlinearity in convection subproblems is explicitly treated by resolving a linear convection system with artificial inflow boundary conditions that can be independently solved. A multistep technique is proposed to rescue the possible instability caused by the explicit treatment of the convection system. Meanwhile, the diffusion subproblems are always self-adjoint and coercive at each time step, and they can be efficiently solved by some existing preconditioned iterative solvers like the preconditioned conjugate galerkin method, and so forth. With the help of finite element discretization, all the major stiffness matrices remain invariant during the time marching process, which makes the present approach extremely fast for the time-dependent nonlinear problems. Finally, several numerical examples are performed to verify the stability, convergence and performance of the new method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1823–1838, 2017

A modified method of approximate particular solutions for solving linear and nonlinear PDEs


The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017

A postprocessed flux conserving finite element solution


We propose a local postprocessing method to get a new finite element solution whose flux is conservative element-wise. First, we use the so-called polynomial preserving recovery (postprocessing) technique to obtain a higher order flux which is continuous across the element boundary. Then, we use special bubble functions, which have a nonzero flux only on one face-edge or face-triangle of each element, to correct the finite element solution element by element, guided by the above super-convergent flux and the element mass. The new finite element solution preserves mass element-wise and retains the quasioptimality in approximation. The method produces a conservative flux, of high-order accuracy, satisfying the constitutive law. Numerical tests in 2D and 3D are presented.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1859–1883, 2017

Approximate solutions of partial differential equations by some Meshfree Greedy Algorithms


In this article, we use some greedy algorithms to avoid the ill-conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well-known PDE examples are solved and compared with the full unsymmetric Kansa's technique. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1884–1899, 2017

An inverse backward problem for degenerate parabolic equations


This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017

Error analysis of mixed finite element method for Poisson-Nernst-Planck system


To improve the convergence rate in L2 norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson-Nernst-Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time-dependent ionic concentrations equations. Optimal error estimates in L∞ ([0,T];L2 ) norm for the electrostatic potential, and in L∞ ([0,T];L2 ) and L∞ ([0,T];H1 ) norms for the ionic concentrations are attained. As a by-product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017

Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method


In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1949–1965, 2017

A partitioned second-order method for magnetohydrodynamic flows at small magnetic reynolds numbers


This article aims to study the partitioned method for magnetohydrodynamic flows at small magnetic Reynolds numbers. We design a partitioned second-order method and show that this method is stable under a time step ( Δ t ) restrict condition. Our method can decouple the magnetohydrodynamic equations so that we can solve two relatively simple subproblems separately at each time step, which is computationally economic. A complete theoretical analysis of error estimates is also given. Finally, we present numerical experiments to support our theory.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1966–1986, 2017

Error bounds of singular boundary method for potential problems


The singular boundary method (SBM) is a recent strong-form boundary collocation method free of integration, mesh, and fictitious boundary. Although an extensive study has been reported in the literature on improving its accuracy and stability as well as its applications to diverse problems, little, however, has been done to analyze its convergence mathematically. The main purpose of this paper is to derive the explicit error bounds of the SBM for potential problems as well as to explain the essential difference between the origin intensity factor (OIF) in the SBM and the singular integration in the boundary element method (BEM). In the process of derivation, we also illustrate the physical meaning of OIF and explain the reason why the OIF has the function to correct the discretization error on the boundary. Finally, several benchmark examples are given to verify the effectiveness of the conclusions obtained from this article, as well as to investigate the different convergence behaviors between the SBM and BEM. It can be found that the SBM has the explicit error bound and is mathematically a stable technique.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1987–2004, 2017

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


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

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation


In this article, we introduce a new, simple, and accurate computational technique for one-dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017

A fast discontinuous finite element discretization for the space-time fractional diffusion-wave equation


In this article, we study fast discontinuous Galerkin finite element methods to solve a space-time fractional diffusion-wave equation. We introduce a piecewise-constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix-vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from O ( N 3 ) required by the traditional methods to O ( N log 2 N ) , where N is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017

An adaptive wavelet space-time SUPG method for hyperbolic conservation laws


This article concerns with incorporating wavelet bases into existing streamline upwind Petrov-Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space-time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems


In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non-symmetric, optimize-then-discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017

Dissipativity of the linearly implicit Euler scheme for Navier-Stokes equations with delay


In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier-Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L2 dissipative for any time step size and H1 dissipative under a time-step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by-product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit-explicit scheme for the celebrated Navier-Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017

High-order compact schemes for fractional differential equations with mixed derivatives


In this article, we consider two-dimensional fractional subdiffusion equations with mixed derivatives. A high-order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes


Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction-correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017