Finite element methods for the Stokes system with interface pressure discontinuities (original) (raw)
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Approximation of the multi-dimensional Stokes system with embedded pressure discontinuities
Advances in Computational Mathematics, 2014
Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present a finite element method to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the Stokes system we study a finite element solution method based on a classical Galerkin-least-squares formulation with an added pressure jump term multiplied by the mesh step size. Both the velocity and the pressure are represented with continuous piecewise linear functions except for the latter field on the embedded interface. 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 this approach.
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 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.
An improved finite element space for discontinuous pressures
Computer Methods in Applied Mechanics and Engineering, 2010
We consider incompressible Stokes flow with an internal interface at which the pressure is discontinuous, as happens for example in problems involving surface tension. We assume that the mesh does not follow the interface, which makes classical interpolation spaces to yield suboptimal convergence rates (typically, the interpolation error in the L 2 (Ω)-norm is of order h 1 2 ).
A divergence-free finite element method for the Stokes problem with boundary correction
Journal of Numerical Mathematics
This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.
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.
Pressure Jump Conditions for Stokes Equations with Discontinuous Viscosity in 2D and 3D
Methods and Applications of Analysis, 2006
In this paper, the jump conditions for the normal derivative of the pressure have been derived for two-phase Stokes (and Navier-Stokes) equations with discontinuous viscosity and singular sources in two and three dimensions. While different jump conditions for the pressure and the velocity can be found in the literature, the jump condition of the normal derivative of the pressure is new. The derivation is based on the idea of the immersed interface method [9, 8] that uses a fixed local coordinate system and the balance of forces along the interface that separates the two phases. The derivation process also provides a way to compute the jump conditions. The jump conditions for the pressure and the velocity are useful in developing accurate numerical methods for two-phase Stokes equations and Navier-Stokes equations.
Journal of Computational and Applied Mathematics, 2008
In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The difficulty arising from the lack of the boundary conditions for the vorticity is overcome by means of a suitable technique for uncoupling both variables. In order to apply the above technique to the Navier-Stokes equations we linearize the advective term in the vorticity transport equation as described in the development of the paper. We illustrate the good performance of our approach by means of numerical results, obtained for benchmark driven cavity problem solved with classical piecewise linear finite element.
A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows
International Journal for Numerical Methods in Fluids, 2012
In this work, a new enrichment space to accomodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared to other existing finite element space and with the classical P 1 space in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with an additional enrichment to accomodate discontinuities in the pressure gradient has also been explored, exhibiting excelent results in problems involving jumps in the density or the volume forces. complicated mathematical structure of the multi-fluid problem, the multi-scale features of the flow, the existence of one or multiple internal interfaces and the unsteadiness of the flow, constitute major challenges for the analysis.
Journal of computational physics, 2006
A numerical method for the simulation of three-dimensional incompressible two-phase flows is presented. The proposed algorithm combines an implicit pressure stabilized finite element method for the solution of incompressible two-phase flow problems with a level set method implemented with a quadrature-free discontinuous Galerkin DG) method [E. Marchandise, J.-F. Remacle, N. Chevaugeon, A quadrature free discontinuous Galerkin method for the level set equation, Journal of Computational Physics 212 (2006) 338-357]. The use of a fast contouring algorithm [N. Chevaugeon, E. Marchandise, C. Geuzaine, J.-F. Remacle, Efficient visualization of high order finite elements, International Journal for Numerical Methods in Engineering] permits us to localize the interface accurately. By doing so, we can compute the discontinuous integrals without neither introducing an interface thickness nor reinitializing the level set. The capability of the resulting algorithm is demonstrated with ''large scale'' numerical examples (free surface flows: dam break, sloshing) and ''small scale'' ones (two phase Poiseuille, Rayleigh-Taylor instability).