A new continuous/discontinuous formulation for Stokes problem with analysis of InfSup stability (original) (raw)
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Stabilized discontinuous finite element approximations for Stokes equations
Journal of Computational and Applied Mathematics, 2007
In this paper, we derive two stabilized discontinuous finite element formulations, symmetric and nonsymmetric, for the Stokes equations and the equations of the linear elasticity for almost incompressible materials. These methods are derived via stabilization of a saddle point system where the continuity of the normal and tangential components of the velocity/displacements are imposed in a weak sense via Lagrange multipliers. For both methods, almost all reasonable pair of discontinuous finite elements spaces can be used to approximate the velocity and the pressure. Optimal error estimate for the approximation of both the velocity of the symmetric formulation and pressure in L 2 norm are obtained, as well as one in a mesh dependent norm for the velocity in both symmetric and nonsymmetric formulations.
Journal of Computational Physics
In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier-Stokes equation which yields the H 1-, H(div)-conforming, and discontinuous Galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness. Under minimum assumption on Galerkin spaces, the semi-and fully-discrete stability is proved when a family of implicit Runge-Kutta methods are used for time discretization. Furthermore, we present a unified discussion on the penalty term. Numerical experiments are presented to compare our schemes with classical schemes in the literature in both unsteady and steady situations. It turns out that our scheme is competitive when applied to well-known benchmark problems such as Taylor-Green vortex, Kovasznay flow, potential flow, lid driven cavity flow, and the flow around a cylinder.
arXiv: Numerical Analysis, 2018
In several studies it has been observed that, when using stabilised mathbbPktimesmathbbPk\mathbb{P}_k^{}\times\mathbb{P}_k^{}mathbbPktimesmathbbPk elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable mathbbPktimesmathbbPk−1\mathbb{P}_k^{}\times\mathbb{P}_{k-1}^{}mathbbPktimesmathbbPk−1 (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not stability the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions.
Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations
International Journal for Numerical Methods in Fluids, 2008
A discontinuous Galerkin (DG) method with solenoidal approximation for the simulation of incompressible flow is proposed. It is applied to the solution of the Stokes equations. The interior penalty method is employed to construct the DG weak form. For every element, the approximation space for the velocity field is decomposed as the direct sum of a solenoidal space and an irrotational space. This allows to split the DG weak form into two uncoupled problems: the first one solves for the velocity and the hybrid pressure (pressure along the mesh edges) and the second one allows the computation of the pressure in the element interior. Furthermore, the introduction of an extra penalty term leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Numerical examples demonstrate the applicability of the proposed methodologies.
Multi-field finite element methods with discontinuous pressures for axisymmetric incompressible flow
Journal of Computational and Applied Mathematics, 2004
Two-and three-ÿeld methods are studied for solving the Stokes system in the axisymmetric case, as a linearized form of di erent types of uid ow equations. Both are designed for the standard Galerkin formulation expressed in terms either of the velocity and the pressure, or of these two ÿelds together with the extra-stress tensor, and use discontinuous pressure spaces. The ÿrst method is related to the rectangular based Q 2 − P 1 element due to Fortin. The other one is linked to the Crouzeix-Raviart triangle. Both methods satisfy the uniform stability (inf-sup) condition relating the velocity and pressure representations, expressed in terms of the natural weighed Sobolev norms, for the system under consideration. This condition is fundamental to derive second-order convergence results for solution methods of viscous or viscoelastic incompressible ow problems based on the corresponding ÿnite element spaces. In order to illustrate this, some numerical results using a method of this type studied by the authors are presented, in connection with the three-ÿeld formulation of the Stokes system related to viscoelastic uids.
Stabilized velocity and pressure mixed hybrid DGFEM for the stokes problem
We propose mixed hybrid Discontinuous Galerkin Finite Element (DGFEM) formulations for the Stokes problem characterized by the introduction of Lagrange multipliers associated with the traces of the velocity and pressure fields on the edges of the elements to weakly impose the transmission conditions. Both velocity and pressure multipliers are stabilized and, as a consequence of these stabilizations, we prove existence and uniqueness of solution for the local problems. All velocity and pressure degrees-of-freedom can be eliminated at the element level by static condensation leading to a global problem in the multipliers only. The proposed methodology is able to recover stability of very convenient choices of finite element spaces, such as those adopting equal order polynomial approximations for all fields. Numerical experiments illustrate the flexibility and robustness of the proposed formulations and show optimal rates of convergence.
Finite element methods for the Stokes system with interface pressure discontinuities
IMA Journal of Numerical Analysis, 2013
Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present two finite element methods to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the two-dimensional Stokes system, we consider solution methods based on piecewise linear approximations of both the velocity and pressure, with either velocity bubble or penalty enrichment, in order to obtain stable discrete problems. Moreover a suitable modification of the pressure space is employed in order to represent interface discontinuities. A priori error analyses point to optimal convergence rates for both approaches, which justify observations from previous numerical experiments carried out in [3].
An interface stabilised finite element method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilising mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. In contrast with discontinuous Galerkin methods, the number of global degrees of freedom is the same as for a continuous method on the same mesh. Different from earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for the appropriate choice of finite element spaces, momentum conservation. Also, in this work a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a point-wise sole...
A hybridizable discontinuous Galerkin method for Stokes flow
Computer Methods in Applied Mechanics and Engineering, 2010
In this paper, we introduce a hybridizable discontinuous Galerkin method for Stokes flow. The method is devised by using the discontinuous Galerkin methodology to discretize a velocity-pressure-gradient formulation of the Stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization. One of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix. Moreover, by using an augmented lagrangian method, the globally coupled unknowns are further reduced to the numerical trace of the velocity only. Another important feature is that the approximations of the velocity, pressure, and gradient converge with the optimal order of k þ 1 in the L 2-norm, when polynomials of degree k P 0 are used to represent the approximate variables. Based on the optimal convergence of the HDG method, we apply an element-by-element postprocessing scheme to obtain a new approximate velocity, which converges with order k þ 2 in the L 2-norm for k P 1. The postprocessing performed at the element level is less expensive than the solution procedure. Numerical results are provided to assess the performance of the method.
The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier-Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior PenaltyGalerkin formulation. The non-linear terms in the Navier-Stokes equations are expressed in the convective form and approximated through the Lesaint-Raviart fluxes modified for DG methods. Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier-Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier-Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier-Stokes solver, we consider unsteady laminar flow pasta square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.