Local projection finite element stabilization for the generalized Stokes problem

We analyze pressure stabilized finite element methods for the solution of the generalized Stokes problem and investigate their stability and convergence properties. An important feature of the method is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations. Although stability of the method has been established, for the homogeneous Stokes equations, the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space. This, makes it a lot simpler and more attractive. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations

Algebraic construction of a third order difference approximations for fractional derivatives and applications

Finite difference approximations for fractional derivatives based on Grunwald formula are well known to be of first order accuracy, but display unstable solutions with known numerical methods. The shifted form of the Grunwald approximation removes this instability and keeps the same first order accuracy. Higher order approximations have been obtained by convex combinations of various shifted Grunwald approximations. Recently, a second order shifted Grunwald type approximation was constructed algebraically through a generating function. In this paper, we derive a new third order approximation from this second order approximation by preconditioning the fractional differential operator. This approximation is used with Crank-Nicolson numerical scheme to approximate the solutions of space-fractional diffusion equations by the same preconditioning. Stability and convergence of the numerical scheme are analysed, supported by numerical results showing third order convergence. References Boris Baeumer, Mihaly Kovacs, and Harish Sankaranarayanan. Higher order grunwald approximations of fractional derivatives and fractional powers of operators. Transactions of the American Mathematical Society, 367(2):813–834, 2015. doi:10.1090/S0002-9947-2014-05887-X E. Barkai, R. Metzler, and J. Klafter. From continuous time random walks to the fractional fokker-planck equation. Physical Review E, 61(1):132, 2000. doi:10.1103/PhysRevE.61.132 Z. Hao, Z. Sun, and W. Cao. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 281:787–805, 2015. doi:10.1016/j.jcp.2014.10.053 Ch Lubich. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3):704–719, 1986. doi:10.1137/0517050 M. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77, 2004. doi:10.1016/j.cam.2004.01.033 H. M. Nasir, B. L. K. Gunawardana, and H. M. N. P. Aberathna. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 3(4):237, 2013. doi:10.7763/IJAPM.2013.V3.212 H. M. Nasir and K. Nafa. A new second order approximation for fractional derivatives with applications. SQU Journal of Science, 23(1):43–55, 2018. doi:10.24200/squjs.vol23iss1pp43-55 W. Tian, H. Zhou, and W. Deng. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 84(294):1703–1727, 2015. doi:10.1090/S0025-5718-2015-02917-2 Y. Yu, W. Deng, and Y. Wu. Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations. arXiv preprint arXiv:1408.6364, 2014. doi:10.4310/CMS.2017.v15.n5.a1 Y. Yu, W. Deng, Y. Wu, and J. Wu. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 112:126–145, 2017. doi:10.1016/j.apnum.2016.10.01 L. Zhao and W. Deng. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 31(5):1345–1381, 2015. doi:10.1002/num.2194

Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

A New Second Order Approximation for Fractional Derivatives with Applications

We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative.  We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results