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    The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes
    (Elsevier Science Sa, 2018) Akbaş, Mine; Linke, Alexander; Rebholz, Leo G.; Schröder, Philipp W.
    grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad-div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix-Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings. (C) 2018 Elsevier B.V. All rights reserved.
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    Analysis of continuous data assimilation scheme for the Navier-Stokes equations using variational multiscale method
    (Elsevier, 2023) Haçat, Gülnur; Akbaş, Mine; Çıbık, Aytekin
    In this study, we analyse a continuous data assimilation (CDA) scheme which enables us to combine an observable data with a numerical method to obtain better solutions in which these solutions are also closely similar to the current state of the system. The scheme is applied on a Navier-Stokes system which is discretized with two-step Backward Differentiation Formula (BDF2) in time and finite element in space. In order to improve the accuracy and prevent some non-physical oscillations due to the effect of small viscosity and the dominance of convection, a projection based variational multiscale method (VMS) has also been applied to the system. We present the long-time stability and long-time convergence analyses of the scheme in details and several numerical tests in order to support theoretical findings and demonstrate the promise of the method.
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    High order algebraic splitting for magnetohydrodynamics simulation
    (Elsevier Science Bv, 2017) Akbaş, Mine; Mohebujjaman, Muhammad; Rebholz, Leo G.; Xiao, Mengying
    This paper proposes, analyzes and tests high order algebraic splitting methods for magnetohydrodynamic (MHD) flows. The main idea is to apply, at each time step, Yosida-type algebraic splitting to a block saddle point problem that arises from a particular incremental formulation of MHD. By doing so, we dramatically reduce the complexity of the nonsymmetric block Schur complement by decoupling it into two Stokes-type Schur complements, each of which is symmetric positive definite and also is the same at each time step. We prove the splitting is 0(Delta t(3)) accurate, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, but also in a finite element context that uses the natural spatial norms. Numerical tests are given to illustrate the theory and show the effectiveness of the method. (C) 2017 Elsevier B.V. All rights reserved.