• Blending low-order stabilised finite element methods: a positivity preserving local projection method for the convection-diffusion equation

      Barrenechea, Gabriel; Burman, Erik; Karakatsani, Fotini; University of Strathclyde; UCL; University of Chester (Elsevier, 2017-01-20)
      In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.
    • Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes

      Barrenechea, Gabriel; Burman, Erik; Karakatsani, Fotini; University of Strathclyde; University College London; University of Chester (Springer, 2016-05-07)
      For the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.
    • A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations

      Karakatsani, Fotini; University of Chester (Oxford University Press, 2015-07-22)
      We derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractional-step #-scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time.
    • A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

      Baensch, Eberhard; Karakatsani, Fotini; Makridakis, Charalambos; University of Erlangen; University of Chester; University of Crete; Foundation for Research & Technology, Greece; University of Sussex (Springer, 2018-05-02)
      This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.