Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time (original) (raw)
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ESAIM: Mathematical Modelling and Numerical Analysis, 2018
In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of theunfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which ...
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Numerische Mathematik, 2013
We consider a time dependent Stokes problem that is motivated by twophase incompressible flow problems with surface tension. The surface tension force results in a right-hand side functional in the momentum equation with poor regularity properties. As a strongly simplified model problem we treat a Stokes problem with a similar time dependent nonsmooth forcing term. We consider the implicit Euler and Crank-Nicolson methods for time discretization. The regularity properties of the data are such that for the Crank-Nicolson method one can not apply error analyses known in the literature. We present a convergence analysis leading to a second order error bound in a suitable negative norm that is weaker that the L 2-norm. Results of numerical experiments are shown that confirm the analysis.
Journal of Computational and Applied Mathematics, 2010
A three-field finite element scheme designed for solving systems of partial differential equations governing time-dependent viscoelastic flows is studied. Once a classical backward Euler time discretization is performed, the resulting three-field system of equations allows for a stable approximation of velocity, pressure and extra stress tensor, by means of continuous piecewise linear finite elements, in both two-and three-dimensional space. This is proved to hold for the linearized form of the system. An advantage of the new formulation is the fact that it provides an algorithm for the explicit iterative resolution of system nonlinearities. Convergence in an appropriate sense applying to these three flow fields is demonstrated.
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International Journal for Numerical Methods in Fluids, 2007
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World Journal of Mechanics, 2011
In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q 1 -P 0 velocity/pressure finite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O( ) in three dimensions. The basic tool in the analysis is the method of modified equations which is applied to finite difference representations of the underlying finite element equations. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system. 2 h
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Optimal Convergence for Time-Dependent Stokes Equation: A New Approach
Journal of Scientific Computing, 2021
In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have suggested a fourth-order compact scheme for the Navier-Stokes equations in streamfunction formulation ∂ t (∆ψ) + (∇ ⊥ ψ) • ∇(∆ψ) = ν∆ 2 ψ. Here we present a new approach for the analysis of a high-order compact scheme for the Navier-Stokes equations in cases where the convective term vanishes, or in cases where the viscous term dominates the convective term. In these cases the Navier-Stokes equations is replaced by the time-dependent Stokes equation ∂ t (∆ψ) = ν∆ 2 ψ. The same type of fourth-order compact schemes that were proposed for the Navier-Stokes equations, may be adopted to the time-dependent Stokes problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence is actually four, thus the error tends to zero as O(h 4), where h is the size of the mesh.
Arbitrary Lagrangian–Eulerian method for Navier–Stokes equations with moving boundaries
Computer Methods in Applied Mechanics and Engineering, 2004
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