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  Subjects -> ENGINEERING (Total: 2428 journals)
    - CHEMICAL ENGINEERING (208 journals)
    - CIVIL ENGINEERING (202 journals)
    - ELECTRICAL ENGINEERING (112 journals)
    - ENGINEERING (1271 journals)
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ENGINEERING (1271 journals)                  1 2 3 4 5 6 7 | Last

Showing 1 - 200 of 1205 Journals sorted alphabetically
3 Biotech     Open Access   (Followers: 8)
3D Research     Hybrid Journal   (Followers: 21)
AAPG Bulletin     Hybrid Journal   (Followers: 8)
AASRI Procedia     Open Access   (Followers: 15)
Abstract and Applied Analysis     Open Access   (Followers: 3)
Aceh International Journal of Science and Technology     Open Access   (Followers: 3)
ACS Nano     Hybrid Journal   (Followers: 278)
Acta Geotechnica     Hybrid Journal   (Followers: 7)
Acta Metallurgica Sinica (English Letters)     Hybrid Journal   (Followers: 7)
Acta Polytechnica : Journal of Advanced Engineering     Open Access   (Followers: 3)
Acta Scientiarum. Technology     Open Access   (Followers: 3)
Acta Universitatis Cibiniensis. Technical Series     Open Access  
Active and Passive Electronic Components     Open Access   (Followers: 7)
Adaptive Behavior     Hybrid Journal   (Followers: 11)
Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi     Open Access  
Adsorption     Hybrid Journal   (Followers: 4)
Advanced Engineering Forum     Full-text available via subscription   (Followers: 7)
Advanced Journal of Graduate Research     Open Access  
Advanced Science     Open Access   (Followers: 5)
Advanced Science Focus     Free   (Followers: 5)
Advanced Science Letters     Full-text available via subscription   (Followers: 10)
Advanced Science, Engineering and Medicine     Partially Free   (Followers: 7)
Advanced Synthesis & Catalysis     Hybrid Journal   (Followers: 17)
Advances in Calculus of Variations     Hybrid Journal   (Followers: 2)
Advances in Catalysis     Full-text available via subscription   (Followers: 5)
Advances in Complex Systems     Hybrid Journal   (Followers: 7)
Advances in Engineering Software     Hybrid Journal   (Followers: 27)
Advances in Fuel Cells     Full-text available via subscription   (Followers: 16)
Advances in Fuzzy Systems     Open Access   (Followers: 5)
Advances in Geosciences (ADGEO)     Open Access   (Followers: 13)
Advances in Heat Transfer     Full-text available via subscription   (Followers: 21)
Advances in Human Factors/Ergonomics     Full-text available via subscription   (Followers: 22)
Advances in Magnetic and Optical Resonance     Full-text available via subscription   (Followers: 9)
Advances in Natural Sciences: Nanoscience and Nanotechnology     Open Access   (Followers: 29)
Advances in Operations Research     Open Access   (Followers: 12)
Advances in OptoElectronics     Open Access   (Followers: 6)
Advances in Physics Theories and Applications     Open Access   (Followers: 13)
Advances in Polymer Science     Hybrid Journal   (Followers: 43)
Advances in Porous Media     Full-text available via subscription   (Followers: 5)
Advances in Remote Sensing     Open Access   (Followers: 44)
Advances in Science and Research (ASR)     Open Access   (Followers: 6)
Aerobiologia     Hybrid Journal   (Followers: 3)
African Journal of Science, Technology, Innovation and Development     Hybrid Journal   (Followers: 6)
AIChE Journal     Hybrid Journal   (Followers: 35)
Ain Shams Engineering Journal     Open Access   (Followers: 5)
Akademik Platform Mühendislik ve Fen Bilimleri Dergisi     Open Access   (Followers: 1)
Alexandria Engineering Journal     Open Access   (Followers: 1)
AMB Express     Open Access   (Followers: 1)
American Journal of Applied Sciences     Open Access   (Followers: 26)
American Journal of Engineering and Applied Sciences     Open Access   (Followers: 10)
American Journal of Engineering Education     Open Access   (Followers: 10)
American Journal of Environmental Engineering     Open Access   (Followers: 16)
American Journal of Industrial and Business Management     Open Access   (Followers: 24)
Analele Universitatii Ovidius Constanta - Seria Chimie     Open Access  
Annals of Combinatorics     Hybrid Journal   (Followers: 4)
Annals of Pure and Applied Logic     Open Access   (Followers: 2)
Annals of Regional Science     Hybrid Journal   (Followers: 7)
Annals of Science     Hybrid Journal   (Followers: 7)
Antarctic Science     Hybrid Journal   (Followers: 1)
Applicable Algebra in Engineering, Communication and Computing     Hybrid Journal   (Followers: 2)
Applicable Analysis: An International Journal     Hybrid Journal   (Followers: 1)
Applied Catalysis A: General     Hybrid Journal   (Followers: 6)
Applied Catalysis B: Environmental     Hybrid Journal   (Followers: 18)
Applied Clay Science     Hybrid Journal   (Followers: 6)
Applied Computational Intelligence and Soft Computing     Open Access   (Followers: 11)
Applied Magnetic Resonance     Hybrid Journal   (Followers: 4)
Applied Nanoscience     Open Access   (Followers: 8)
Applied Network Science     Open Access   (Followers: 3)
Applied Numerical Mathematics     Hybrid Journal   (Followers: 5)
Applied Physics Research     Open Access   (Followers: 5)
Applied Sciences     Open Access   (Followers: 3)
Applied Spatial Analysis and Policy     Hybrid Journal   (Followers: 5)
Arabian Journal for Science and Engineering     Hybrid Journal   (Followers: 5)
Archives of Computational Methods in Engineering     Hybrid Journal   (Followers: 5)
Archives of Foundry Engineering     Open Access  
Archives of Thermodynamics     Open Access   (Followers: 8)
Arkiv för Matematik     Hybrid Journal   (Followers: 1)
ASEE Prism     Full-text available via subscription   (Followers: 3)
Asia-Pacific Journal of Science and Technology     Open Access  
Asian Engineering Review     Open Access  
Asian Journal of Applied Science and Engineering     Open Access   (Followers: 1)
Asian Journal of Applied Sciences     Open Access   (Followers: 2)
Asian Journal of Biotechnology     Open Access   (Followers: 9)
Asian Journal of Control     Hybrid Journal  
Asian Journal of Current Engineering & Maths     Open Access  
Asian Journal of Technology Innovation     Hybrid Journal   (Followers: 8)
Assembly Automation     Hybrid Journal   (Followers: 2)
at - Automatisierungstechnik     Hybrid Journal   (Followers: 1)
ATZagenda     Hybrid Journal  
ATZextra worldwide     Hybrid Journal  
Australasian Physical & Engineering Sciences in Medicine     Hybrid Journal   (Followers: 1)
Australian Journal of Multi-Disciplinary Engineering     Full-text available via subscription   (Followers: 2)
Autonomous Mental Development, IEEE Transactions on     Hybrid Journal   (Followers: 9)
Avances en Ciencias e Ingeniería     Open Access  
Balkan Region Conference on Engineering and Business Education     Open Access   (Followers: 1)
Bangladesh Journal of Scientific and Industrial Research     Open Access  
Basin Research     Hybrid Journal   (Followers: 5)
Batteries     Open Access   (Followers: 6)
Bautechnik     Hybrid Journal   (Followers: 1)
Bell Labs Technical Journal     Hybrid Journal   (Followers: 28)
Beni-Suef University Journal of Basic and Applied Sciences     Open Access   (Followers: 4)
BER : Manufacturing Survey : Full Survey     Full-text available via subscription   (Followers: 1)
BER : Motor Trade Survey     Full-text available via subscription  
BER : Retail Sector Survey     Full-text available via subscription   (Followers: 1)
BER : Retail Survey : Full Survey     Full-text available via subscription   (Followers: 1)
BER : Survey of Business Conditions in Manufacturing : An Executive Summary     Full-text available via subscription   (Followers: 3)
BER : Survey of Business Conditions in Retail : An Executive Summary     Full-text available via subscription   (Followers: 3)
Beyond : Undergraduate Research Journal     Open Access  
Bhakti Persada : Jurnal Aplikasi IPTEKS     Open Access  
Bharatiya Vaigyanik evam Audyogik Anusandhan Patrika (BVAAP)     Open Access   (Followers: 1)
Bilge International Journal of Science and Technology Research     Open Access  
Biofuels Engineering     Open Access   (Followers: 1)
Biointerphases     Open Access   (Followers: 1)
Biomaterials Science     Full-text available via subscription   (Followers: 11)
Biomedical Engineering     Hybrid Journal   (Followers: 15)
Biomedical Engineering and Computational Biology     Open Access   (Followers: 13)
Biomedical Engineering Letters     Hybrid Journal   (Followers: 5)
Biomedical Engineering, IEEE Reviews in     Full-text available via subscription   (Followers: 21)
Biomedical Engineering, IEEE Transactions on     Hybrid Journal   (Followers: 37)
Biomedical Engineering: Applications, Basis and Communications     Hybrid Journal   (Followers: 5)
Biomedical Microdevices     Hybrid Journal   (Followers: 8)
Biomedical Science and Engineering     Open Access   (Followers: 4)
Biomedizinische Technik - Biomedical Engineering     Hybrid Journal   (Followers: 1)
Biomicrofluidics     Open Access   (Followers: 4)
BioNanoMaterials     Hybrid Journal   (Followers: 2)
Biotechnology Progress     Hybrid Journal   (Followers: 40)
Bitlis Eren University Journal of Science and Technology     Open Access  
Boletin Cientifico Tecnico INIMET     Open Access  
Botswana Journal of Technology     Full-text available via subscription   (Followers: 1)
Boundary Value Problems     Open Access   (Followers: 1)
Brazilian Journal of Science and Technology     Open Access   (Followers: 2)
Broadcasting, IEEE Transactions on     Hybrid Journal   (Followers: 12)
Bulletin of Canadian Petroleum Geology     Full-text available via subscription   (Followers: 13)
Bulletin of Engineering Geology and the Environment     Hybrid Journal   (Followers: 14)
Bulletin of the Crimean Astrophysical Observatory     Hybrid Journal  
Cahiers, Droit, Sciences et Technologies     Open Access  
Calphad     Hybrid Journal   (Followers: 2)
Canadian Geotechnical Journal     Hybrid Journal   (Followers: 31)
Canadian Journal of Remote Sensing     Full-text available via subscription   (Followers: 42)
Case Studies in Engineering Failure Analysis     Open Access   (Followers: 6)
Case Studies in Thermal Engineering     Open Access   (Followers: 5)
Catalysis Communications     Hybrid Journal   (Followers: 6)
Catalysis Letters     Hybrid Journal   (Followers: 2)
Catalysis Reviews: Science and Engineering     Hybrid Journal   (Followers: 7)
Catalysis Science and Technology     Free   (Followers: 8)
Catalysis Surveys from Asia     Hybrid Journal   (Followers: 3)
Catalysis Today     Hybrid Journal   (Followers: 7)
CEAS Space Journal     Hybrid Journal   (Followers: 2)
Cellular and Molecular Neurobiology     Hybrid Journal   (Followers: 3)
Central European Journal of Engineering     Hybrid Journal  
Chaos : An Interdisciplinary Journal of Nonlinear Science     Hybrid Journal   (Followers: 2)
Chaos, Solitons & Fractals     Hybrid Journal   (Followers: 3)
Chinese Journal of Catalysis     Full-text available via subscription   (Followers: 2)
Chinese Journal of Engineering     Open Access   (Followers: 2)
Chinese Science Bulletin     Open Access   (Followers: 1)
Ciencia e Ingenieria Neogranadina     Open Access  
Ciencia en su PC     Open Access   (Followers: 1)
Ciencias Holguin     Open Access   (Followers: 3)
CienciaUAT     Open Access   (Followers: 1)
Cientifica     Open Access  
CIRP Annals - Manufacturing Technology     Full-text available via subscription   (Followers: 11)
CIRP Journal of Manufacturing Science and Technology     Full-text available via subscription   (Followers: 13)
City, Culture and Society     Hybrid Journal   (Followers: 22)
Clay Minerals     Full-text available via subscription   (Followers: 10)
Clean Air Journal     Full-text available via subscription   (Followers: 1)
Clinical Science     Full-text available via subscription   (Followers: 9)
Coal Science and Technology     Full-text available via subscription   (Followers: 3)
Coastal Engineering     Hybrid Journal   (Followers: 11)
Coastal Engineering Journal     Hybrid Journal   (Followers: 6)
Coatings     Open Access   (Followers: 4)
Cogent Engineering     Open Access   (Followers: 2)
Cognitive Computation     Hybrid Journal   (Followers: 4)
Color Research & Application     Hybrid Journal   (Followers: 2)
COMBINATORICA     Hybrid Journal  
Combustion Theory and Modelling     Hybrid Journal   (Followers: 14)
Combustion, Explosion, and Shock Waves     Hybrid Journal   (Followers: 14)
Communications Engineer     Hybrid Journal   (Followers: 1)
Communications in Numerical Methods in Engineering     Hybrid Journal   (Followers: 2)
Components, Packaging and Manufacturing Technology, IEEE Transactions on     Hybrid Journal   (Followers: 28)
Composite Interfaces     Hybrid Journal   (Followers: 7)
Composite Structures     Hybrid Journal   (Followers: 283)
Composites Part A : Applied Science and Manufacturing     Hybrid Journal   (Followers: 211)
Composites Part B : Engineering     Hybrid Journal   (Followers: 251)
Composites Science and Technology     Hybrid Journal   (Followers: 195)
Comptes Rendus Mécanique     Full-text available via subscription   (Followers: 2)
Computation     Open Access   (Followers: 1)
Computational Geosciences     Hybrid Journal   (Followers: 16)
Computational Optimization and Applications     Hybrid Journal   (Followers: 7)
Computational Science and Discovery     Full-text available via subscription   (Followers: 2)
Computer Applications in Engineering Education     Hybrid Journal   (Followers: 8)
Computer Science and Engineering     Open Access   (Followers: 20)
Computers & Geosciences     Hybrid Journal   (Followers: 31)
Computers & Mathematics with Applications     Full-text available via subscription   (Followers: 8)
Computers and Electronics in Agriculture     Hybrid Journal   (Followers: 5)
Computers and Geotechnics     Hybrid Journal   (Followers: 11)
Computing and Visualization in Science     Hybrid Journal   (Followers: 7)
Computing in Science & Engineering     Full-text available via subscription   (Followers: 34)
Conciencia Tecnologica     Open Access  
Concurrent Engineering     Hybrid Journal   (Followers: 3)
Continuum Mechanics and Thermodynamics     Hybrid Journal   (Followers: 8)

        1 2 3 4 5 6 7 | Last

Journal Cover
Applied Numerical Mathematics
Journal Prestige (SJR): 0.93
Citation Impact (citeScore): 1
Number of Followers: 5  
 
  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274
Published by Elsevier Homepage  [3163 journals]
  • Localized kernel-based approximation for pricing financial options under
           regime switching jump diffusion model
    • Abstract: Publication date: Available online 18 July 2018Source: Applied Numerical MathematicsAuthor(s): Reza Mollapourasl, Majid Haghi, Ruihua Liu In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.
       
  • A second-order, uniquely solvable, energy stable BDF numerical scheme for
           the phase field crystal model
    • Abstract: Publication date: Available online 18 July 2018Source: Applied Numerical MathematicsAuthor(s): Qi Li, Liquan Mei, Bo You In this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme.
       
  • A fast numerical method for two-dimensional Riesz space fractional
           diffusion equations on a convex bounded region
    • Abstract: Publication date: Available online 17 July 2018Source: Applied Numerical MathematicsAuthor(s): S. Chen, F. Liu, I. Turner, V. Anh Fractional differential equations have attracted considerable attention because of their many applications in physics, geology, biology, chemistry, and finance. In this paper, a two-dimensional Riesz space fractional diffusion equation on a convex bounded region (2D-RSFDE-CBR) is considered. These regions are more general than rectangle or circular domains. A novel alternating direction implicit method for the 2D-RSFDE-CBR with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the method are discussed. The resulting linear systems are Toeplitz-like and are solved by the preconditioned conjugate gradient method with a suitable circulant preconditioner. By the fast Fourier transform, the method only requires a computational cost of O(nlog⁡n) per time step. These numerical techniques are used for simulating a two-dimensional Riesz space fractional FitzHugh-Nagumo model. The numerical results demonstrate the effectiveness of the method. These techniques can be extended to three spatial dimensions, which will be the topic of our future research.
       
  • A novel finite difference discrete scheme for the time fractional
           diffusion-wave equation
    • Abstract: Publication date: Available online 11 July 2018Source: Applied Numerical MathematicsAuthor(s): Zhengguang Liu, Aijie Cheng, Xiaoli Li In this article, we consider initial and boundary value problems for the diffusion-wave equation involving a Caputo fractional derivative(of order α, with 1
       
  • Numerical analysis of a collocation method for functional integral
           equations
    • Abstract: Publication date: Available online 10 July 2018Source: Applied Numerical MathematicsAuthor(s): Adson M. Rocha, Juarez S. Azevedo, Saulo P. Oliveira, Maicon R. Correa We study the numerical approximation of functional integral equations, a class of nonlinear Fredholm-type integral equations of the second kind, by the collocation method with piecewise continuous basis functions. The resulting nonlinear algebraic system is solved with the Picard iteration method. Starting from the analysis of the continuous problem in L∞([a,b]), we prove the convergence of numerical solution and, under an additional regularity assumption, provide an a priori error estimate. Numerical examples illustrate the predicted theoretical results.
       
  • The lumped mass FEM for a time-fractional cable equation
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Mariam Al-Maskari, Samir Karaa We consider the numerical approximation of a time-fractional cable equation involving two Riemann–Liouville fractional derivatives. We investigate a semidiscrete scheme based on the lumped mass Galerkin finite element method (FEM), using piecewise linear functions. We establish optimal error estimates for smooth and middly smooth initial data, i.e., v∈Hq(Ω)∩H01(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L2(Ω), the optimal L2(Ω)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L∞(Ω)-norm error estimate is also obtained. Further, we analyze two fully discrete schemes using convolution quadrature in time based on the backward Euler and the second-order backward difference methods, and derive error estimates for smooth and nonsmooth data. Finally, we present several numerical examples to confirm our theoretical results.
       
  • High order compact finite difference schemes on nonuniform grids
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): R. Čiegis, O. Suboč In this paper we consider high-order compact finite difference schemes constructed on 1D non-uniform grids. We apply them to parabolic and Schrödinger equations. Stability of these schemes is investigated by using the spectral method. Computer experiments are applied in order to find critical grids for which the stability condition is violated. Such grids are obtained for the Schrödinger problem, but not for the parabolic problems. Numerical examples supporting our theoretical analysis are provided and discussed.
       
  • Numerical simulation of flow in smectic liquid crystals
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Merlin Fallahpour, Sean McKee, Ewa B. Weinmüller Our aim is to simulate a nonlinear system of ODEs describing the flow in smectic liquid crystals. The nonlinear system is first linearized. We present a direct approach to compute the exact analytic solution of this linear system and use this solution as a starting profile in the Matlab package bvpsuite2.0 to obtain the approximate solution to the nonlinear system. Although, the solution of the nonlinear system has steep boundary layers and therefore is difficult to resolve, we demonstrate that bvpsuite2.0 can cope with the problem and provide an approximation with reasonable accuracy.
       
  • On the new properties of Caputo–Fabrizio operator and its application in
           deriving shifted Legendre operational matrix
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Jian Rong Loh, Abdulnasir Isah, Chang Phang, Yoke Teng Toh In this paper, we study the recently introduced Caputo and Fabrizio operator, which this new operator was derived by replacing the singular kernel in the classical Caputo derivative with the regular kernel. We introduce some useful properties based on the definition by Caputo and Fabrizio for a general order n
       
  • Modulus-based matrix splitting algorithms for the quasi-complementarity
           problems
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Shi-Liang Wu, Peng Guo In this paper, a class of modulus-based matrix splitting iteration methods for the quasi-complementarity problems is presented. The convergence analysis of the proposed methods is discussed. Numerical experiments show that the proposed methods are efficient.
       
  • NURBS curves in direct definition of the shapes of the boundary for 2D
           Stokes flow problems in modified classical BIE
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Eugeniusz Zieniuk, Krzysztof Szerszeń This paper proposes a modeling of the boundary at the interface between the body and the fluid for 2D flow problems described by the Stokes equation with NURBS curves. The theoretical representation of the boundary with the help of these curves is directly included into the classical boundary integral equations (BIE) for the Stokes equation. After this analytical inclusion of NURBS curves into a mathematical formula of the classical BIE, new generalized parametric integral systems (PIES) are obtained.
       
  • Smooth subgrid fields underpin rigorous closure in spatial discretisation
           of reaction–advection–diffusion PDEs
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): G.A. Jarrad, A.J. Roberts Finite difference/element/volume methods of spatially discretising pdes impose a subgrid scale interpolation on the dynamics. In contrast, the so-called holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is systematically informed by the underlying microscale dynamics. We establish a new proof that there exists an exact closure of the spatially-discrete dynamics of a general class of reaction–advection–diffusion pdes. The approach also constructs new systematic approximations to the in-principle closure starting from a basis of simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the constructed holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers' pde, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to empower systematic construction of good, macroscale discretisations to a wide range of dissipative and wave pdes.
       
  • Two-grid methods for expanded mixed finite element approximations of
           semi-linear parabolic integro-differential equations
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Tianliang Hou, Luoping Chen, Yin Yang In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.
       
  • A spectral framework for fractional variational problems based on
           fractional Jacobi functions
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): M.A. Zaky, E.H. Doha, J.A. Tenreiro Machado A family of orthogonal systems of fractional functions is introduced. The proposed orthogonal systems are based on Jacobi polynomials through a fractional coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of fractional differential models. Approximation errors by the basic orthogonal projection are established. Three new kinds of fractional Jacobi–Gauss-type interpolations are introduced. As an example of application, an efficient approximation based on the proposed fractional functions to a fractional variational problem is presented and implemented. This approximation takes into account the potential irregularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. Implementation details are provided for the scheme, together with a series of numerical examples to show the efficiency of the proposed method.
       
  • Construction and comparison of multidimensional spectral variational
           integrators and spectral collocation methods
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Yiqun Li, Boying Wu, Melvin Leok In this paper, we construct numerical schemes for spectral collocation methods and spectral variational integrators which converge geometrically. We present a systematic comparison of how spectral collocation methods and Galerkin spectral variational integrators perform in terms of their ability to reproduce accurate trajectories in configuration and phase space, their ability to conserve momentum and energy, as well as the linear stability of these methods when applied to some classical Hamiltonian systems.
       
  • An inexact Newton-like conditional gradient method for constrained
           nonlinear systems
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): M.L.N. Gonçalves, F.R. Oliveira In this paper, we propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose derivatives satisfy a Hölder-like condition and the other is for functions that satisfy a Smale condition, which includes a substantial class of analytic functions. Some preliminary numerical experiments illustrating the applicability of the proposed method are also presented.
       
  • A stable enriched Galerkin element for the Stokes problem
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Nabil Chaabane, Vivette Girault, Beatrice Riviere, Travis Thompson We propose a stable element for the divergence operator that approximates the velocity by continuous linear polynomials plus piecewise constants and the pressure by piecewise constants. A uniform inf–sup condition is obtained for conforming meshes in two or three dimensions. The resulting method belongs to the class of enriched Galerkin methods, and is applied to the solution of a Stokes system. A priori error estimates in the energy norm and in the L2 norm are derived. Extensions to the Navier–Stokes system are presented.
       
  • An adaptive step size controller for iterative implicit methods
    • Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Lukas Einkemmer The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used.In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion–advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen–Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.
       
  • Error propagation for implicit–explicit general linear methods
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): M. Braś, A. Cardone, Z. Jackiewicz, P. Pierzchała We consider the class of implicit–explicit general linear methods (IMEX). Such schemes are designed for ordinary differential equation systems with right hand side function splitted into stiff and non-stiff parts. We investigate error propagation of IMEX methods up to the terms of order p+2. In addition, we construct IMEX schemes of order p and stage order q=p, p≤4 and we verify the performance of methods in several numerical experiments.
       
  • An efficient technique based on finite difference/finite element method
           for solution of two-dimensional space/multi-time fractional Bloch–Torrey
           equations
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Mehdi Dehghan, Mostafa Abbaszadeh The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6]. The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O(τ2−α). Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains.
       
  • An effective numerical method for solving fractional pantograph
           differential equations using modification of hat functions
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): S. Nemati, P. Lima, S. Sedaghat In this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of convergence order O(h3).
       
  • The Williams step increases the stability and accuracy of the hoRA time
           filter
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Ahmet Guzel, Catalin Trenchea The explicit weakly-stable second-order accurate leapfrog scheme is widely used in the numerical models of weather and climate, in conjunction with the Robert–Asselin (RA) and Robert–Asselin–Williams (RAW) time filters. The RA and RAW filters successfully suppress the spurious computational mode associated with the leapfrog method, but also weakly damp the physical mode and degrade the numerical accuracy to first-order. The recent higher-order Robert–Asselin (hoRA) time filter reduces the undesired numerical damping of the RA and RAW filters and increases the accuracy to second up-to third-order. We prove that the combination of leapfrog-hoRA and Williams' step increases the stability by 25%, improves the accuracy of the amplitude of the physical mode up-to two significant digits, effectively suppresses the computational modes, and further diminishes the numerical damping of the hoRA filter.
       
  • The approximate solution of nonlinear Volterra integral equations of the
           second kind using radial basis functions
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Pouria Assari, Mehdi Dehghan In this investigation, a computational scheme is given to solve nonlinear one- and two-dimensional Volterra integral equations of the second kind. We utilize the radial basis functions (RBFs) constructed on scattered points by combining the discrete collocation method to estimate the solution of Volterra integral equations. All integrals appeared in the scheme are approximately computed by the composite Gauss–Legendre integration formula. The implication of previous methods for solving these types of integral equations encounters difficulties by increasing the dimensional of problems and sometimes requires a mesh generation over the solution region. While the new technique presented in the current paper does not increase the difficulties for higher dimensional integral equations due to the easy adaption of RBF and also needs no cell structures on the domains. Moreover, we obtain the error bound and the convergence rate of the proposed approach. Illustrative examples clearly show the reliability and efficiency of the method and confirm the theoretical error estimates.
       
  • A block-centered finite difference method for the distributed-order
           time-fractional diffusion-wave equation
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Xiaoli Li, Hongxing Rui In this article, a block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation with Neumann boundary condition is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δt1+σ/2+h2+k2+σ2) both for pressure and velocity are established on non-uniform rectangular grids, where Δt,h,k and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
       
  • Superconvergence analysis of a new low order nonconforming MFEM for
           time-fractional diffusion equation
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Dongyang Shi, Huaijun Yang A new low order nonconforming mixed finite element method (MFEM) is proposed and analyzed for time-fractional diffusion equation with element pair (CNRQ1+Q0×Q0). A new error estimate for the consistency error of nonconforming element CNRQ1 is proved, which leads to the superclose and superconvergence results of the original variable in broken H1 norm, and of the flux in L2 norm for a fully-discrete scheme with the Caputo derivative approximated by the classical L1 method. The results obtained herein improve the corresponding conclusions in the previous literature. Finally, some numerical results are provided to confirm the theoretical analysis.
       
  • Weak Milstein scheme without commutativity condition and its error bound
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Toshihiro Yamada This paper shows a discretization method of solution to stochastic differential equations as an extension of the Milstein scheme. With a simple method, we reconstruct weak Milstein scheme through second order polynomials of Brownian motions without assuming the Lie bracket commutativity condition on vector fields imposed in the classical Milstein scheme and show a sharp error bound for it. Numerical example illustrates the validity of the scheme.
       
  • Sixth-order non-uniform combined compact difference scheme for multi-term
           time fractional diffusion-wave equation
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Z. Soori, A. Aminataei In this paper, we propose a high-order scheme for the numerical solution of multi-term time fractional diffusion-wave (FDW) equation in one and two-dimensional on non-uniform grids. Based on the sixth-order non-uniform combined compact difference (NCCD) scheme in the space directions on non-uniform grids, an alternating direction implicit (ADI) method is proposed to split the equation into two separate one dimensional equations. The multi-term time fractional derivation is described in the Caputo's sense with scheme of order O(τ3−α), 1
       
  • Wave dynamics on networks: Method and application to the sine-Gordon
           equation
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Denys Dutykh, Jean-Guy Caputo We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.
       
  • Error estimates of a trigonometric integrator sine pseudo-spectral method
           for the extended Fisher–Kolmogorov equation
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Xin Li, Luming Zhang In this article, a trigonometric integrator sine pseudo-spectral (TISP) method is presented for the extended Fisher–Kolmogorov equation. This method depends on a Gautschi-type integrator in phase space to the temporal approximation and the sine pseudo-spectral method to the spatial discretization. Rigorous error estimates are carried out in the energy space by utilizing the mathematical induction. The error bound shows the new scheme which established by the TISP method has second-order accurate in time and spectral-order accurate in space. Moreover, the new scheme is generalized to higher dimensions. The compact finite difference (CFD) scheme in one and two dimensions which supported by the method of order reduction are constructed as a benchmark for comparisons. Comparison results between two schemes are given to confirm the theoretical studies and demonstrate the efficiency and accuracy of TISP method in both one and multi-dimensional problems.
       
  • Convergence estimates for multigrid algorithms with SSC smoothers and
           applications to overlapping domain decomposition
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): E. Aulisa, G. Bornia, S. Calandrini, G. Capodaglio In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs under no regularity assumptions on the solution of the problem. The proposed analysis provides three main contributions to the existing theory. The first novel contribution of this study is a convergence bound that depends on the number of multigrid smoothing iterations. This result is obtained under no regularity assumptions on the solution of the problem. A similar result has been shown in the literature for the cases of full regularity and partial regularity assumptions. Second, our theory applies to local refinement applications with arbitrary level hanging nodes. More specifically, for the smoothing algorithm we provide subspace decompositions that are suitable for applications where the multigrid spaces are defined on finite element grids with arbitrary level hanging nodes. Third, global smoothing is employed on the entire multigrid space with hanging nodes. When hanging nodes are present, existing multigrid strategies advise to carry out the smoothing procedure only on a subspace of the multigrid space that does not contain hanging nodes. However, with such an approach, if the number of smoothing iterations is increased, convergence can improve only up to a saturation value. Global smoothing guarantees an arbitrary improvement in the convergence when the number of smoothing iterations is increased. Numerical results are also included to support our theoretical findings.
       
  • The spectral-Galerkin approximation of nonlinear eigenvalue problems
    • Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Jing An, Jie Shen, Zhimin Zhang In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form −div(A∇u)+Vu+f(u2)u=λu,‖u‖L2=1. We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.
       
  • Estimates of the discrete van Cittert deconvolution error in approximate
           deconvolution models of turbulence in bounded domains
    • Abstract: Publication date: Available online 4 July 2018Source: Applied Numerical MathematicsAuthor(s): Argus A. Dunca Recent turbulence models such as the Approximate Deconvolution Model (ADM) or the Leray-deconvolution model are derived from the Navier–Stokes equations using the van Cittert approximate deconvolution method. As a consequence, the numerical error in the approximation of the Navier–Stokes weak solution with discrete solutions of the above models is influenced also by the discrete deconvolution error u−DNu‾h caused by the approximate deconvolution method. Here u is the flow field, u‾ is its average and DN is the N-th order van Cittert deconvolution operator.It is therefore important to analyze the deconvolution error u−DNu‾h in terms of the mesh size h, the filter radius α and the order N of the deconvolution operators used in the computation.This problem is investigated herein in the case of bounded domains and zero-Dirichlet boundary conditions. It is proved that on a sequence of quasiuniform meshes the L2 norm of the discrete deconvolution error convergences to 0 in the order of hk+1+KNh provided that the filter radius is in the order of the mesh size and the flow field has enough regularity. Here K
       
  • Fourth order wave equation with nonlinear strain and logarithmic
           nonlinearity
    • Abstract: Publication date: Available online 27 June 2018Source: Applied Numerical MathematicsAuthor(s): Runzhang Xu, Wei Lian, Xiangkun Kong, Yanbing Yang The main goal of this work is to investigate the initial boundary value problem of fourth order wave equation with nonlinear strain and logarithmic nonlinearity at three different initial energy levels, i.e., subcritical energy E(0)d. First, we prove the local existence of weak solution by Galerkin method. In the framework of potential well, we obtain the global existence and infinite time blow up of the solution with sub-critical initial energy. Moreover by the scaling technique, we obtain global existence and infinite time blow up of the solution with critical initial energy. Also, a high energy infinite time blow up result is established.
       
  • Global existence and blow up of the solution for nonlinear Klein–Gordon
           equation with general power-type nonlinearities at three initial energy
           levels
    • Abstract: Publication date: Available online 6 June 2018Source: Applied Numerical MathematicsAuthor(s): Yongbing Luo, Yanbing Yang, Md Salik Ahmed, Tao Yu, Mingyou Zhang, Ligang Wang, Huichao Xu This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein–Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.
       
  • Manakov model with gain/loss terms and N-soliton interactions: Effects of
           periodic potentials
    • Abstract: Publication date: Available online 30 May 2018Source: Applied Numerical MathematicsAuthor(s): V.S. Gerdjikov, M.D. Todorov We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the perturbed nonlinear Schrödinger equation (NLSE) and the Manakov model. The perturbations include the simultaneous by a periodic external potential, and linear and nonlinear gain/loss terms. We derive the corresponding perturbed complex Toda chain (PCTC) models for both NLSE and Manakov model. We show that the soliton interactions dynamics for the PCTC models compares favorably to full numerical results of the original perturbed NLSE and Manakov model.
       
  • Stability band structure for periodic states in periodic potentials
    • Abstract: Publication date: Available online 18 May 2018Source: Applied Numerical MathematicsAuthor(s): Bin Liu, Lu Li, Boris A. Malomed A class of periodic solutions of the nonlinear Schrödinger equation with non-Hermitian potentials are considered. The system may be implemented in planar nonlinear optical waveguides carrying an appropriate distribution of local gain and loss, in a combination with a photonic-crystal structure. The complex potential is built as a solution of the inverse problem, which predicts the potential supporting required periodic solutions. The main subject of the analysis is the spectral structure of the linear (in)stability for the stationary spatially periodic states in the periodic potentials. The stability and instability bands are calculated by means of the plane-wave-expansion method, and verified in direct simulations of the perturbed evolution. The results show that the periodic solutions may be stable against perturbations in specific Floquet–Bloch bands, even if they are unstable against small random perturbations.
       
  • Spectral stability of travelling wave solutions in a Keller–Segel
           model
    • Abstract: Publication date: Available online 17 May 2018Source: Applied Numerical MathematicsAuthor(s): P.N. Davis, P. van Heijster, R. Marangell We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise.
       
  • Long wave approximation using conformal mapping for large-amplitude
           internal waves in a two-fluid system
    • Abstract: Publication date: Available online 11 April 2018Source: Applied Numerical MathematicsAuthor(s): Sunao Murashige This paper describes a new type of long wave model for periodic internal waves propagating in permanent form at the interface between two immiscible inviscid fluids. This model for irrotational plane motion of these waves is derived in the complex velocity potential planes where the flow domains are conformally mapped. Since no smallness assumption of wave amplitude is made and the wave elevation at the interface is represented by a single-valued function of the velocity potential, this model is applicable to large-amplitude motions of which wave profile may overhang. Numerical examples demonstrate that the proposed model can produce overhanging solutions, and variations of solutions with wavelength or wave amplitude are qualitatively similar to those of the full Euler system. It is also pointed out that the kinematic condition at the interface is exactly satisfied in the proposed model for all wave amplitudes, but not in an existing long wave model derived in the physical plane.
       
  • Error analysis of finite element approximations of the optimal control
           problem for stochastic Stokes equations with additive white noise
    • Abstract: Publication date: Available online 5 March 2018Source: Applied Numerical MathematicsAuthor(s): Youngmi Choi, Hyung-Chun Lee Finite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. To obtain the most efficient deterministic optimal control, we set up the cost functional as we proposed in [20]. Error estimates are established for the fully coupled optimality system using Green's functions and Brezzi–Rappaz–Raviart theory. Numerical examples are also presented to examine our theoretical results.
       
  • An adaptive nonlinear elimination preconditioned inexact Newton algorithm
           for highly local nonlinear multicomponent PDE systems
    • Abstract: Publication date: Available online 12 February 2018Source: Applied Numerical MathematicsAuthor(s): Haijian Yang, Feng-Nan Hwang This work aims to develop an adaptive nonlinear elimination preconditioned inexact Newton method as the numerical solution of large sparse multi-component partial differential equation systems with highly local nonlinearity. A nonlinear elimination algorithm used as a nonlinear preconditioner has been shown to be a practical technique for enhancing the robustness and improving the efficiency of an inexact Newton method for some challenging problems, such as the transonic full potential problems. The basic idea of our method is to remove some components causing troubles in order to decrease the impact of local nonlinearity on the global system. The two key elements of the method are the valid identification of the to-be-eliminated components and the choice of subspace correction systems, respectively. In the method, we employ the point-wise residual component of nonlinear systems as an indicator for selecting these to-be-eliminated components adaptively and build a subspace nonlinear system consisting of the components corresponding to the bad region and an auxiliary linearized subsystem to reduce the interfacial jump pollution. The numerical results demonstrate that the new approach significantly improves performance for incompressible fluid flow and heat transfer problems with highly local nonlinearity when compared to the classical inexact Newton method.
       
  • On a parallel, 3-dimensional, finite element solver for viscous,
           resistive, stationary magnetohydrodynamics equations: Velocity–current
           formulation
    • Abstract: Publication date: Available online 3 February 2018Source: Applied Numerical MathematicsAuthor(s): K.D. Brauss, A.J. Meir We describe a parallel implementation for the numerical approximation of solutions to the three-dimensional viscous, resistive magnetohydrodynamics (MHD) equations using a velocity–current formulation. In comparison to other formulations, the velocity–current formulation presented in this paper is an integro-differential system of equations that incorporates nonideal boundaries and nonlinearities due to induction. The solution to the equations is approximated using a Picard iteration, discretized with the finite element method, and solved iteratively with the Krylov subspace method GMRES. Effective preconditioning strategies are required to numerically solve the resulting equations with Krylov solvers [12]. For GMRES convergence, the system matrix resulting from the discretization of the velocity–current formulation is preconditioned using a simple, block-diagonal Schur-complement preconditioner based on [14]. The MHD solver is implemented using freely available, well-documented, open-source, libraries deal.II, p4est, Trilinos, and PETSc, capable of scaling to tens of thousands of processors on state-of-the-art HPC architectures.
       
  • Optimal error estimates of both coupled and two-grid decoupled methods for
           a mixed Stokes–Stokes model
    • Abstract: Publication date: Available online 31 January 2018Source: Applied Numerical MathematicsAuthor(s): Yuhong Zhang, Haibiao Zheng, Yanren Hou, Li Shan In this paper, we provide a coupled algorithm and a two-grid decoupled algorithm for a mixed Stokes–Stokes model, which is coupled by a nonlinear interface transmission condition. The coupled algorithm is to discretize the mixed model directly by standard finite element method. For the two-grid decoupled algorithm, we first solve the mixed model on a coarse grid, and update the solution on a fine grid by two separated Stokes problems. Under a hypothesis about the regularity of analytical solutions, optimal error estimates for two algorithms are achieved. Several numerical tests are given to verify our theoretical results.
       
 
 
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