4,303 research outputs found

    Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

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    We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

    Spectral/hp element methods: recent developments, applications, and perspectives

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    The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

    Stabilisation of high aspect ratio mixed finite elements for incompressible flow

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    Anisotropically refined mixed finite elements are beneficial for the resolution of local features such as boundary layers. Unfortunately, the stability of the resulting scheme is highly sensitive to the aspect ratio of the elements. Previous analysis revealed that the degeneration arises from a relatively small number of spurious (piecewise constant) pressure modes. The present article is concerned with resolving the problem of how to suppress the spurious pressure modes in order to restore stability yet at the same time not incur any deterioration in the approximation properties of the reduced pressure space. Two results are presented. The first gives the minimal constraints on the pressure space needed to restore stability with respect to aspect ratio and shows that the approximation properties of the constrained pressure space and the unconstrained pressure space are essentially identical. Alternatively, one can impose the constraint weakly through the use of a stabilized finite element scheme. A second result shows that the stabilized finite element scheme is robust with respect to the aspect ratio of the elements and produces an approximation that satisfies an error bound of the same type to the mixed finite element scheme using the constrained space

    Mixed hp‐DGFEM for incompressible flows II: Geometric edge meshes

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    We consider the Stokes problem of incompressible fluid flow in three‐dimensional polyhedral domains discretized on hexahedral meshes with hp‐discontinuous Galerkin finite elements of type Qk for the velocity and Qk−1 for the pressure. We prove that these elements are inf‐sup stable on geometric edge meshes that are refined anisotropically and non‐quasiuniformly towards edges and corners. The discrete inf‐sup constant is shown to be independent of the aspect ratio of the anisotropic elements and is of O(k−3/2) in the polynomial degree k, as in the case of conforming Qk−Qk−2 approximations on the same meshe

    h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

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    In this work we exploit agglomeration based hh-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature hh-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L2L^2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.Comment: 78 pages, 7 figure

    Adaptive finite element simulation of incompressible flows by hybrid continuous-discontinuous Galerkin formulations

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    In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes. Then we develop an equal-order stabilized finite element formulation for incompressible flows over these hybrid spaces, which combines the element interior stabilization of SUPG-type continuous Galerkin formulations and the jump stabilization of discontinuous Galerkin formulations. Optimal stability and convergence results are obtained. For the adaptive setting, we use a standard error estimator and marking strategy. Numerical experiments show the optimal accuracy of the hybrid algorithm for both uniformly and adaptively refined nonconforming meshes. The outcome of this work is a finite element formulation that can naturally be used on nonconforming meshes, as discontinuous Galerkin formulations, while keeping the much lower CPU cost of continuous Galerkin formulations
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