Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations

Abstract This paper provides a Lagrange nodal discontinuous Galerkin method for solving the time-dependent incompressible space fractional Navier-Stokes equations numerically. The existence and uniqueness of weak solutions are obtained. By combining the Lagrange method in temporal discretization and the hybridized discontinuous Galerkin method in spatial direction, the fully discrete scheme is presented and the stability is proved rigorously. Furthermore, the error estimates for the L2-norm are derived in both the velocity and the pressure. Finally, some numerical experiments are given to illustrate the performance of the proposed method and validate the theoretical result.

[1]  Yvon Maday,et al.  A high-order characteristics/finite element method for the incompressible Navier-Stokes equations , 1997 .

[2]  Weihua Deng,et al.  Central local discontinuous Galerkin method for the space fractional diffusion equation , 2018, Comput. Math. Appl..

[3]  M. Ahmadinia,et al.  Analysis of local discontinuous Galerkin method for time–space fractional convection–diffusion equations , 2018 .

[4]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[5]  Xu Chuan-ju,et al.  A FRACTIONAL STOKES EQUATION AND ITS SPECTRAL APPROXIMATION , 2017 .

[6]  Haijin Wang,et al.  Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow , 2017, Math. Comput..

[7]  Nicholas Hale,et al.  An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations , 2012, SIAM J. Sci. Comput..

[8]  A. Priestley,et al.  Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature , 1994 .

[9]  Masahisa Tabata,et al.  An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates , 2017, Math. Comput..

[10]  L. Richardson,et al.  Atmospheric Diffusion Shown on a Distance-Neighbour Graph , 1926 .

[11]  Endre Süli,et al.  Stability of the Lagrange-Galerkin method with non-exact integration , 1988 .

[12]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[13]  Shuming Liu,et al.  Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation , 2019, Communications on Applied Mathematics and Computation.

[14]  Jiwei Zhang,et al.  Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations , 2018, SIAM J. Numer. Anal..

[15]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[16]  Zhiyong Si.  Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems , 2011, Numerical Algorithms.

[17]  I. Podlubny.  Fractional differential equations , 1998 .

[18]  Aiguo Xiao,et al.  An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions , 2018, Numerical Algorithms.

[19]  Meng Li,et al.  Galerkin finite element method for nonlinear fractional Schrödinger equations , 2017, Numerical Algorithms.

[20]  Weihua Deng,et al.  A Hybridized Discontinuous Galerkin Method for 2D Fractional Convection–Diffusion Equations , 2015, Journal of Scientific Computing.

[21]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[22]  Jiahong Wu,et al.  Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces , 2006 .

[23]  Jiye Yang,et al.  Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations , 2015, J. Comput. Phys..

[24]  Huazhong Tang,et al.  High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations , 2012 .

[25]  Paul Castillo,et al.  On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method , 2018, J. Sci. Comput..

[26]  Weihua Deng,et al.  Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes , 2014, J. Comput. Phys..

[27]  Wenchang Tan,et al.  Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics , 2006 .

[28]  Well-posedness for fractional Navier-Stokes equations in critical spaces close to $\dot{B}^{-(2\beta-1)}_{\infty,\infty}(\mathbb{R}^{n})$ , 2009, 0906.5140.

[29]  S. Holm,et al.  Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. , 2004, The Journal of the Acoustical Society of America.

[30]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[31]  Endre Süli,et al.  Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations , 1988 .

[32]  Jianfei Huang,et al.  Finite element method for two-dimensional space-fractional advection-dispersion equations , 2015, Appl. Math. Comput..

[33]  G. Karniadakis,et al.  A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations , 2016 .

[34]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[35]  Jan S. Hesthaven,et al.  Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations , 2013, SIAM J. Numer. Anal..

[36]  Tomasz W. Dłotko,et al.  Fractional Navier-Stokes equations , 2017 .

[37]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[38]  J. Hesthaven,et al.  Local discontinuous Galerkin methods for fractional diffusion equations , 2013 .

[39]  Chengjian Zhang,et al.  Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions , 2016, Comput. Math. Appl..

[40]  Zhiping Mao,et al.  A Spectral Penalty Method for Two-Sided Fractional Differential Equations with General Boundary Conditions , 2018, SIAM J. Sci. Comput..

[41]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[42]  Guido Kanschat,et al.  Local Discontinuous Galerkin Methods for the Stokes System , 2002, SIAM J. Numer. Anal..