9 research outputs found
A Pressure-Robust Discretization of Oseen's Equation Using Stabilization in the Vorticity Equation
Discretization of Navier--Stokes equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an error estimate in the -norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based streamline upwind Petrov--Galerkin stabilization
Analysis of a stabilised finite element method for power-law fluids
A low-order finite element method is constructed and analysed for an
incompressible non-Newtonian flow problem with power-law rheology. The method
is based on a continuous piecewise linear approximation of the velocity field
and piecewise constant approximation of the pressure. Stabilisation, in the
form of pressure jumps, is added to the formulation to compensate for the
failure of the inf-sup condition, and using an appropriate lifting of the
pressure jumps a divergence-free approximation to the velocity field is built
and included in the discretisation of the convection term. This construction
allows us to prove the convergence of the resulting finite element method for
the entire range r>\frac{2 d}{d+2} of the power-law index for which weak
solutions to the model are known to exist in space dimensions,
On Basic Iteration Schemes for Nonlinear AFC Discretizations
Algebraic flux correction (AFC) finite element discretizations of steady-state convection-diffusion-reaction equations lead to a nonlinear problem. This paper presents first steps of a systematic study of solvers for these problems. Two basic fixed point iterations and a formal Newton method are considered. It turns out that the fixed point iterations behave often quite differently. Using a sparse direct solver for the linear problems, one of them exploits the fact that only one matrix factorization is needed to become very efficient in the case of convergence. For the behavior of the formal Newton method, a clear picture is not yet obtained
A unified analysis of algebraic flux correction schemes for convection–diffusion equations
Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?
On the Use of Anisotropic Triangles with Mixed Finite Elements: Application to an “Immersed” Approach for Incompressible Flow Problems
In this chapter, we discuss the use of some common mixed finite elements in the context of a locally anisotropic remeshing strategy, close in philosophy to "immersed" approaches for interface problems. A characteristic of the present method is the presence of highly flat triangles. Such a distinctive feature may imply stability issues for mixed elements with incompressible flow problems. First, we present a review of the literature dealing with interface problems and we illustrate these results with a simple 1D framework alongside of numerical tests. Second, we present the locally anisotropic remeshing approach for interface problems in 2D with a focus on the incompressible Stokes problem. We then present numerical tests to show stability issues of common mixed elements, as well as possible stable ones. We also deal with conditioning issues. Finally, we illustrate the results with two applications, including the fluid-structure interaction of a rotational rigid bar