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This paper introduces a finite volume element method for solving the time-tempered fractional advection–diffusion equation with weakly singular solution at initial time t = 0 . An innovative approach is proposed to construct an intelligent non-uniform temporal mesh, which significantly reduces errors as compared to using a uniform temporal mesh. The error reduction is quantified in terms of percentage improvement of errors. Due to the presence of a large number of integral calculations involving complicated functions, we used parallel computing techniques to accelerate the computation process. The stability of the method is rigorously proven, and numerical examples are provided to demonstrate the effectiveness of the method and validate the theoretical results. [ABSTRACT FROM AUTHOR]

The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence. [ABSTRACT FROM AUTHOR]

In this article, our aim is to consider an efficient finite volume element method combined with the L 2 − 1 σ formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the L 2 − 1 σ formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result. [ABSTRACT FROM AUTHOR]

In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers' equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the L 1 formula and approximating the spatial terms using FVEM. The convergence of the scheme is proven. Finally, numerical examples are provided to confirm the scheme's validity. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.

In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in L 2 (J ; L 2) and L ∞ (J ; L 2) -norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is O (h 3 / 2 + △ t) in the sense of L 2 (J ; L 2) and L ∞ (J ; L 2) -norm. Finally, a numerical experiment is presented to test these theoretical results. [ABSTRACT FROM AUTHOR]

We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle. We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O (∥ q ∥ ∞ h + ∥ g ∥ ∞ h 2) -acute and h is small enough, where h denotes the mesh size, q and g are coefficients of the convection and reaction terms, respectively. To deal with the convection-dominated problems, we use the upwind triangle technique. For such scheme, the mesh condition can be sharper to O (∥ g ∥ ∞ h 2) -acute. Some numerical examples are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we solve a second-order nonlinear elliptic equation by using the finite volume element method, and give the rigorous error estimates. Firstly, the computational domain is divided into general convex quadrilateral meshes. We choose the isoparametric bilinear element space as the trial function space and the piecewise constant function space as the test function space, and construct the corresponding finite volume element scheme. Secondly, on the h 2 -parallelogram mesh, the boundedness and coercivity of bilinear form are proved. Using the Brouwer fixed point theorem, we give the existence and uniqueness of numerical solution. Thirdly, we derive the estimates of ‖ ∇ (u − u h) ‖ L t with t ≥ 2 and ‖ u − u h ‖ 0 under certain regularity assumptions. At last, we carry out numerical experiments on quadrilateral meshes and calculate the convergence orders in H 1 and L 2 norms, which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, a fast second-order finite volume element (FVE) algorithm is proposed to solve the nonlinear time-fractional coupled diffusion model based on the time two-mesh (TT-M) computing method. In this algorithm, the integer and Riemann-Liouville fractional derivatives are approximated by the second-order backward difference formula and the WSGD formula respectively, the time interval is divided into coarse and fine meshes, then the three steps TT-M FVE algorithm is constructed by using the interpolation operator. The existence and uniqueness for the TT-M FVE algorithm are analyzed in detail, the asymptotically optimal a priori error estimates for variables u and v in the discrete L ∞ (L 2 (Ω)) and L2(H1(Ω) norms are obtained. It is shown that when time coarse and fine mesh sizes satisfy τ c = O (τ f 1 / 2) , the fast algorithm can achieve the same accuracy as the FVE algorithm, and reduce more computational cost. Finally, some numerical results are given to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical L 1 -formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in L 2 (Ω) -norm are derived, but it is difficult to obtain the corresponding results in H 1 (Ω) -norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness. [ABSTRACT FROM AUTHOR]

The paper proposes a hybrid numerical method for solving a model problem of two-phase nonequilibrium flow of an incompressible fluid in a porous medium. This problem is relevant in the modern theory of the motion of multiphase fluids in porous media and has many applications. The studied model is based on the assumption that the relative phase permeabilities and capillary pressure depend not only on saturation, but also on its time derivative. The saturation equation in this problem refers to the type of convection-diffusion with a predominance of convection, which also includes a third-order term to account for the nonequilibrium effects. Due to the hyperbolic nature of the equation, its solution is accompanied by a number of difficulties that lead to the need for an appropriate choice of the solution method. In contrast to previous works, this paper uses a finite volume element method for solving the problem, the construction of which is based on integral balance equations, and an approximate solution is chosen from the finite element space. To discretize the problem, two different dual grids are used based on the main triangulation. In this paper, a number of a priori estimates are obtained which yields the unconditional stability of the scheme as well as its convergence with the second order. The advantages of the approach used include the local conservatism of the scheme, as well as the comparative simplicity of the software implementation of the method. These results are confirmed by a numerical test carried out on the example of a model problem.

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we propose a finite volume element method of primal-dual type to solve the ill-posed elliptic problem, that is, the elliptic problem with lacking or overlapping boundary value condition. We first establish the primal-dual finite volume element scheme by introducing the Lagrange multiplier λ and prove the well-posedness of the discrete scheme. Then, the error estimations of the finite volume solution are derived under some proper norms including the H 1 -norm. Numerical experiments are provided to verify the effectiveness of the proposed finite volume element method at last. [ABSTRACT FROM AUTHOR]

Abstract We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples.

In this paper, several domain decomposition algorithms are considered to solve parallelly non-steady diffusion equations. The contributions of this work are threefold. The first one is that we present effective parallel domain decomposition schemes underlying the classical and monotone finite volume element discretization. These schemes adopt a three-stage technique, which is also used in the difference method and the finite volume method. Positivity of the parallel scheme based on monotone finite volume element method is proved. The second one is that we propose a new domain decomposition computational strategy to improve the correction step in prediction-correction frame such that the whole parallel process is simplified. The third one is that we consider a parallel algorithm build on quadratic finite volume element method, which preserves local conservation and has higher convergence rates. These parallel schemes are of intrinsic parallelism since we apply the prediction-correction technique to the interface of subdomains coming from an artificial division of computational domain or multi-medium physical domain. The proposed parallel schemes need only local communication among neighboring processors. Numerical results are presented to illuminate the accuracy, stability and parallelism of the parallel schemes. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems. [ABSTRACT FROM AUTHOR]

Abstract In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I h ∗ $I_{h}^{*}$ , a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L ∞ ( L 2 ( Ω ) ) $L^{\infty }(L^{2}(\varOmega))$ and L 2 ( H 1 ( Ω ) ) $L^{2}(H^{1}(\varOmega))$ norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness.

One difficulty in solving the Poisson–Nernst–Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convection‐dominant problem in the Nernst–Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady‐state PNP equations is proposed and analyzed. Numerical tests on four ion‐channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable. A new stabilized technique for the mixed boundary condition on the interface is proposed and analyzed.The new stabilized PNP solver is fulfilled in Fortran and Python, and well validated using the test models.The effectiveness of the new stabilized technique as well as the robustness of the new PNP solver is demonstrated. [ABSTRACT FROM AUTHOR]

We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples. [ABSTRACT FROM AUTHOR]

Poisson-Boltzmann equation (PBE) is a classic implicit continuum model to predict the electrostatic potentials of a solvated biomolecule. In this paper, we present a finite volume element method specific to the elliptic interface problem with a non-homogeneous ux condition for solving PBE and provide a follow-up analysis. The new PBE solver is fulfilled through both Fortran and Python, afterwards the local Poisson test model coupled with an analytical solution is adopted to well validate the program. Lastly, an application of the new solver to the prediction of solvation free energies of the proteins is made. [ABSTRACT FROM AUTHOR]

In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

• The method is easy to implement, and has good conservation properties. • The scheme is efficient, only needs to solve a linear equations at each time step. • The scheme is robust, and suitable for long time computation. • The scheme can be naturally extended to multi-dimensional problems. In this paper, a new linearized energy-preserving Crank-Nicolson finite volume element scheme is derived for the improved Boussinesq equation. The fully discrete scheme can be shown to conserve both mass and energy in the discrete setting. It is proved that the scheme is uniquely solvable and convergent with the rate of order two in a discrete L 2 norm. At last, a series of numerical experiments on typical improved Boussinesq and Sine–Gordon equations are provided to verify our theoretical results and to show the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order P 1 − P 0 element for the Stokes region and P 1 element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the H 1 -norm for the velocity and piezometric head and in the L 2 -norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment. [ABSTRACT FROM AUTHOR]

In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I h ∗ , a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L ∞ (L 2 (Ω)) and L 2 (H 1 (Ω)) norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness. [ABSTRACT FROM AUTHOR]

The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

In order to effectively improve the ventilation performance of permanent magnet synchronous traction motor used in the rail transit，this paper，taking a 600 kW permanent magnet synchronous traction motor as an example，and based on computational fluid dynamics ( CFD) and the basic theory of heat transfer，establishes the three dimensional physical model under its ventilation scheme for the analysis of temperature field. Applying the finite volume element method，each part of the permanent magnet synchronous traction motor was numerical analyzed. And the temperature distribution of its internal components was summarized in order to optimizing. It provides the theory basis forimproving the permanent magnet traction motor cooling performance as well as the accurate calculation for temperature field of the permanent magnet traction motor with larger capacity.

Abstract Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we provide a new type of study approach for the two-dimensional (2D) Sobolev equations. We first establish a semi-discrete Crank-Nicolson (CN) formulation with second-order accuracy about time for the 2D Sobolev equations. Then we directly establish a fully discrete CN finite volume element (CNFVE) formulation from the semi-discrete CN formulation about time and provide the error estimates for the fully discrete CNFVE solutions. Finally, we provide a numerical example to verify the correction of theoretical conclusions. Further, it is shown that the fully discrete CNFVE formulation is better than the fully discrete FVE formulation with first-order accuracy in time.

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the L1-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator Ih*. The unconditional stability result and the optimal a priori error estimate in the L2(Ω)-norm for the Crank–Nicolson FVE scheme are obtained by using the direct recursive method. Finally, some numerical results are given to verify the time and space convergence accuracy, and to examine the feasibility and effectiveness for the proposed scheme.

In this paper, we present a stabilized finite volume element method with the conforming finite element triples P 1 – P 0 – P 1 and P 1 – P 1 – P 1 for approximating the velocity, pressure, and hydraulic head of a coupled Stokes–Darcy problem. The proposed method is convenient to implement, computationally efficient, mass conserving, optimally accurate, and able to handle complex geometries; therefore, this method has great potential to be useful for realistic problems involving coupled free flow and porous media flow. To offset the lack of the inf-sup condition of the P 1 – P 0 and P 1 – P 1 elements for the Stokes equation, a parameter free stabilization term is added to the discrete formulation. Stability and optimal error estimates are proved based on a bridge built up between the finite volume element method and the finite element method. An element level implementation of the stabilization term is discussed so that an existing code package can be conveniently modified to handle the stabilization procedures. A series of numerical experiments are provided to illustrate the above features of the proposed method, the theoretical results, and the realistic applications. [ABSTRACT FROM AUTHOR]

In this paper, a linear implicit energy-preserving and a linear explicit momentum-preserving finite volume element scheme are proposed for the Gardner equation. The proposed schemes are derived by using the discrete variational derivative method (DVDM) in time and the finite volume element method (FVEM) in space. The conservative properties and the linear stability of the proposed schemes are analyzed. In particular, the proposed methods are compared with a nonlinear Crank-Nicolson FVEM and a third-order Runge-Kutta FVEM in terms of accuracy, CPU time and conservative properties. [ABSTRACT FROM AUTHOR]

The numerical solution for the time fractional advection‐diffusion problem in one‐dimension with the initial‐boundary condition is proposed in this paper by B‐spline finite volume element method. The fractional derivative is Caputo in the proposed scheme. The stability of the proposed numerical method is studied, and the numerical results presented support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in H 1 -norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation. [ABSTRACT FROM AUTHOR]

In this paper, an energy-preserving finite volume element scheme is proposed for the Korteweg-de Vries equation. The scheme is a combination of the discrete variational derivative method in time and the finite volume element method in space. The scheme can precisely conserve the global mass and energy at the discrete level, as well as has higher accuracy. For comparison, we also propose a momentum-preserving scheme and a finite volume element scheme. The numerical results demonstrate the remarkable accuracy and efficiency of our method compared with other schemes. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, residual-type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual-type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual-type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259-275, 2017 [ABSTRACT FROM AUTHOR]

In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg-de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the spatially semi-discrete piecewise linear finite volume element method for the time fractional sub-diffusion problem in two dimensions, and give an approximate solution of this problem. At first, we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods. Furthermore, we use the standard finite element Ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Finally, some numerical examples are included to illustrate the effectiveness of the new technique. [ABSTRACT FROM AUTHOR]

In this article, a Crank-Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O( h2 + k2) is proved for the numerical solution of the control, state and adjoint-state in a discrete L2-norm. To derive this result, we also get the error estimate (convergent order O( h2 + k2)) of Crank-Nicolson finite volume element approximation for the second-order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331-1356, 2016 [ABSTRACT FROM AUTHOR]

In this paper, we provide a new type of study approach for the two-dimensional (2D) Sobolev equations. We first establish a semi-discrete Crank-Nicolson (CN) formulation with second-order accuracy about time for the 2D Sobolev equations. Then we directly establish a fully discrete CN finite volume element (CNFVE) formulation from the semi-discrete CN formulation about time and provide the error estimates for the fully discrete CNFVE solutions. Finally, we provide a numerical example to verify the correction of theoretical conclusions. Further, it is shown that the fully discrete CNFVE formulation is better than the fully discrete FVE formulation with first-order accuracy in time. [ABSTRACT FROM AUTHOR]

In this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid sizeH), and the fine grid solution (with grid sizeh) can be obtained in a single symmetric and linear step. The optimalH1error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfyh= 𝒪(H3|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation. [ABSTRACT FROM PUBLISHER]

An integral equation approach is proposed to solve the unsteady convection–diffusion equations. In this approach, the second order Adams–Moulton method is firstly utilized for the time discretization. Then by using the Green's function of the Laplace equation in the series form, the convection–diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system. The accuracy, convergence and the stability of this integral equation approach are examined by four examples. In comparison with the characteristic variational multiscale method and the finite volume element method, the integral equation approach shows a higher accuracy. Compared with the finite volume element method the integral equation approach has a better convergence. In solving the convection dominated convection–diffusion problems the integral equation approach demonstrates a good stability. [ABSTRACT FROM AUTHOR]