Stabilized velocity and pressure mixed hybrid DGFEM for the stokes problem (original) (raw)
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Naturally stable or stabilized finite element formulations to solve incompressible flow problems governed by Stokes equations have been the goal of research for a long time. Mixed finite element methods for the pressure and the velocity fields with different stabilization forms have been widely used. Recently, hybrid formulations have been developed associated with Discontinuous Galerkin methods. In this context, we propose dual hybridized mixed formulations for the Stokes problem, characterized by the introduction of the Lagrange multipliers associated with velocity or velocity and pressure fields to weakly impose continuity on each edge of the elements. The proposed methodology is able to recover stability of very convenient finite element spaces, such as those adopting equal order polynomial approximations. Numerical experiments illustrate the flexibility and robustness of the proposed formulations and show optimal rates of convergence.
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.
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.
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.
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.
The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow
SIAM Journal on Numerical Analysis, 2009
In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.
ETNA - Electronic Transactions on Numerical Analysis
In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for the Poisson and Stokes problems in 2D, considering only piecewise Lagrangian finite elements. The complication here lies in the fact that the classical Babuška-Brezzi theory is difficult to verify for low-order finite elements, so we proceed in a non-standard way. First, we prove uniqueness, and then we apply a discrete version of Fredholm's alternative theorem to ensure existence. The a-priori error analysis is done by introducing suitable projections of the exact solution. As a result, we prove that the method is convergent, and, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed.
A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method.
ESAIM: Mathematical Modelling and Numerical Analysis
We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.
Hybridizable Discontinuous Galerkin Methods
Lecture notes in computational science and engineering, 2010
In this paper, we present and discuss the so-called hybridizable discontinuous Galerkin (HDG) methods. The discontinuous Galerkin (DG) methods were originally devised for numerically solving linear and then nonlinear hyperbolic problems. Their success prompted their extension to the compressible Navier-Stokes equations-and hence to second-order elliptic equations. The clash between the DG methods and decades-old, well-established finite element methods resulted in the introduction of the HDG methods. The HDG methods can be implemented more e ciently and are more accurate than all previously known DG methods; they represent a competitive alternative to the well established finite element methods. Here we show how to devise and implement the HDG methods, argue why they work so well and prove optimal convergence properties in the framework of di↵usion and incompressible flow problems. We end by briefly describing extensions to other continuum mechanics and fluid dynamics problems.