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Author: Novo, Julia ×

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    On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows
    (Amsterdam [u.a.] : Elsevier Science, 2021) García-Archilla, Bosco; John, Volker; Novo, Julia
    The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.
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    An adaptive SUPG method for evolutionary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) de Frutos, Javier; García-Archilla, Bosco; John, Volker; Novo, Julia
    An adaptive algorithm for the numerical simulation of time-dependent convectiondiffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convectiondiffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6]
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    On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) John, Volker; Novo, Julia
    Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.
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    Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Frutos, Javier de; García-Archilla, Bosco; John, Volker; Novo, Julia
    The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and CrankNicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results
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    Projection methods for incompressible flow problems with WENO finite difference schemes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Frutos, Javier de; John, Volker; Novo, Julia
    Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convectiondiffusion equations [?]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several nonincremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.
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    Analysis of the PSPG stabilization for the continuous-in-time discretization of the evolutionary stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) John, Volker; Novo, Julia
    Optimal error estimates for the pressure stabilized Petrov-Galerkin (PSPG) method for the continuous-in-time discretization of the evolutionary Stokes equations are proved in the case of regular solutions. The main result is applicable to higher order finite elements. The error bounds for the pressure depend on the error of the pressure at the initial time. An approach is suggested for choosing the discrete initial velocity in such a way that this error is bounded. The "instability of the discrete pressure for small time steps", which is reported in the literature, is discussed on the basis of the analytical results. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.
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    Finite element methods for the incompressible Stokes equations with variable viscosity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) John, Volker; Kaiser, Kristine; Novo, Julia
    Finite element error estimates are derived for the incompressible Stokes equations with variable viscosity. The ratio of the supremum and the infimum of the viscosity appears in the error bounds. Numerical studies show that this ratio can be observed sometimes. However, often the numerical results show a weaker dependency on the viscosity.
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    A robust SUPG norm a posteriori error estimator for the SUPG finite element approximation of stationary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) John, Volker; Novo, Julia
    A robust residual-based a posteriori estimator is proposed for the SUPG finite element method applied to stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of this paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypotheses. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided. 1
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    Error analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) John, Volker; Moreau, Baptiste; Novo, Julia
    A reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov--Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. Numerical studies illustrate the performance of the SUPG-ROM method.
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    Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) John, Volker; Knobloch, Petr; Novo, Julia
    The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed.