Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

John, Volker; Knobloch, Petr; Novo, Julia

doi:10.1007/s00791-018-0290-5

Cited by 71 publications

(64 citation statements)

References 139 publications

(187 reference statements)

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“…We consider the time-dependent incompressible Navier-Stokes equations [66,61,27] There are references regarding the historical development of finite element methods (FEM) for the Navier-Stokes problem (1) until 2016; cf., for example, the monograph [39]. A summary of very recent results for H H H 1 -conforming FEM, together with several open problems, can be found in the review paper [40].…”

Section: Introductionmentioning

confidence: 99%

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Schroeder

Lehrenfeld

Linke

et al. 2018

SeMA

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Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption ∇u ∈ L 1 (0, T ; L ∞ (Ω )) which is discussed in detail. In the sense of best practice, we review and establish pressure-and Re-semi-robust estimates for pointwise divergence-free H 1 -conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.Keywords time-dependent incompressible flow · Re-semi-robust error estimates · pressure-robustness · inf-sup stable methods · exactly divergence-free FEMA relatively new aspect in the FE analysis applied to incompressible flows is 'pressurerobustness' [41]. In its most general form, pressure-robustness of a numerical method is defined by its ability to fulfil the following requirement: if the exact solution u u u of (1) belongs to the approximation space V V V h , i.e. if u u u ∈ V V V h , then the discrete solution u u u h coincides with the exact one, that is, u u u h = u u u. In certain physical regimes of the incompressible Navier-Stokes equations -i.e., in certain benchmarks -pressure-robustness allows to use less expensive discretisation schemes without losing accuracy [49,1]. As a consequence, the following fundamental invariance principle transfers from the continuous level to the discretised case: Replacing the source term f f f by f f f + ∇ψ changes the solution (u u u, p) to (u u u, p + ψ). For example, in a potential flow, (u u u · · · ∇)u u u can be very large but it is a gradient and therefore balanced by the pressure gradient and thus does not have any impact on the velocity field. Only recently it has been shown that high Reynolds number potential flows are really challenging for the numerical solution with standard low-order Galerkin-FEM [50,41].A well-known important consequence for methods which are not pressure-robust is that already for the steady incompressible Stokes problem the velocity error estimates for kinetic and dissipation energies are corrupted by the pressure approximability multiplied by ν −1/2 [41,49]. Note that the mechanism responsible for the excitation of this kind of numerical error is a completely linear phenomenon. Exactly divergence-free FEM are naturally pressure-robust, but classical inf-sup stable velocity-pressure pairs like Taylor-Hood FEM are usually not pressure-robust. In fact, ...

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Section: Introductionmentioning

confidence: 99%

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Schroeder

Lehrenfeld

Linke

et al. 2018

SeMA

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“…In the subsequent passage from SODE to difference schemes (when time derivatives are replaced by differences) the resulting difference schemes become implicit due to the non-diagonality of the matrix M . In addition, the matrix M turns out to be time-dependent in some statements of numerical problems, which can lead to the necessity to inverse/factorize the matrix M at each time step of integration of the SODE obtained [4][5][6][7]. The mass lumping technique [1][2][3][4][7][8][9][10][11] is often used in computational practice to facilitate computational efforts and avoid the necessity for sophisticated (and computationally high-cost) algebra.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, owing to Lemma 2 we have ) In accordance with the above results (see Lemmas 1-2 and Remarks 1-2), the signs of . Recall that the convectiondiffusion ratio is usually characterized by the so-called Peclet number[1,2,[5][6][7], which is an important physical parameter that shows the rate of the prevalence of convection over diffusion. Lemma 1 states that for an arbitrary finite Peclet number of n , see Propositions 3 and 5 for details.…”

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confidence: 99%

A note on the application of the Guermond–Pasquetti mass lumping correction technique for convection–diffusion problems

Siryk

2019

Journal of Computational Physics

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We provide a careful Fourier analysis of the Guermond-Pasquetti mass lumping correction technique [Guermond J.-L., Pasquetti R. A correction technique for the dispersive effects of mass lumping for transport problems // Computer Methods in Applied Mechanics and Engineering. -2013. -Vol. 253. -P. 186-198] applied to pure transport and convection-diffusion problems. In particular, it is found that increasing the number of corrections reduces the accuracy for problems with diffusion; however all the corrected schemes are more accurate than the consistent Galerkin formulation in this case. For the pure transport problems the situation is the opposite. We also investigate the differences between two numerical solutionsthe consistent solution and the corrected ones, and show that increasing the number of corrections makes solutions of the corrected schemes closer to the consistent solution in all cases.

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“…The aspect ratio affects the accuracy and stability of the numerical solution [55]. Given the flat elements, and that analysis and stabilization parameter choices for anisotropic meshes is still a relatively unexplored corner of numerical analysis [42], it is important to consider the definition of the cell size h K in τ GLS in Eq. 10.…”

Section: The Cell Size H Kmentioning

confidence: 99%

“…Ice-sheet simulations involve the solution of the p-Stokes equations on very flat domains (i.e., on anisotropic meshes), with complex boundaries and boundary conditions. In a recent review study, anisotropic meshes and realistic boundary conditions for incompressible flow problems, were identified as two of the most important at large open problems of finite element research [42], without overlooking important works of, e.g., [2,48]. The authors also called for more systematic studies or stabilized methods and choices of stablization parameters for practical simulations.…”

Section: Introductionmentioning

confidence: 99%

Stabilized equal low-order finite elements in ice sheet modeling – accuracy and robustness

Helanow

Ahlkrona

2018

Comput Geosci

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We investigate the accuracy and robustness of one of the most common methods used in glaciology for finite element discretization of the p-Stokes equations: linear equal order finite elements with Galerkin least-squares (GLS) stabilization on anisotropic meshes. Furthermore, we compare the results to other stabilized methods. We find that the vertical velocity component is more sensitive to the choice of GLS stabilization parameter than horizontal velocity. Additionally, the accuracy of the vertical velocity component is especially important since errors in this component can cause ice surface instabilities and propagate into future ice volume predictions. If the element cell size is set to the minimum edge length and the stabilization parameter is allowed to vary non-linearly with viscosity, the GLS stabilization parameter found in literature is a good choice on simple domains. However, near ice margins the standard parameter choice may result in significant oscillations in the vertical component of the surface velocity. For these reasons, other stabilization techniques, in particular the interior penalty method, result in better accuracy and are less sensitive to the choice of stabilization parameter. During this work, we also discovered that the manufactured solutions often used to evaluate errors in glaciology are not reliable due to high artificial surface forces at singularities. We perform our numerical experiments in both FEniCS and Elmer/Ice.

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Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

Cited by 71 publications

References 139 publications

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

A note on the application of the Guermond–Pasquetti mass lumping correction technique for convection–diffusion problems

Stabilized equal low-order finite elements in ice sheet modeling – accuracy and robustness

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