An adaptive Uzawa FEM for the Stokes problem: Convergence without the inf-sup condition

We introduce and study an adaptive finite element method (FEM) for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity, whereas for pressure the elements can be either discontinuous of degree k - 1 or continuous of degree k -1 and k. The popular Taylor-Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi-optimal.Fil: Bänsch, Eberhard. Freie Universität Berlin;Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Nochetto, Ricardo Horacio. University of Maryland; Estados Unido

A posteriori error control for fully discrete Crank–Nicolson schemes

We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank--Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, which is introduced in the present paper

Surface diffusion of graphs: Variational formulation, error analysis, and simulation

Surface diffusion is a (fourth-order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for graphs and derive a priori error estimates for a time-continuous finite element discretization. We also introduce a semi-implicit time discretization and a Schur complement approach to solve the resulting fully discrete, linear systems. After computational verification of the orders of convergence for polynomial degrees 1 and 2, we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion, as well as the onset of singularities in finite time, such as infinite slopes and cracks.Fil: Bänsch, Eberhard. Freie Universität Berlin; . Weierstrass Institute For Applied Analysis And Stochastics;Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Nochetto, Ricardo Horacio. University of Maryland; Estados Unido

Quasi-stability of the primary flow in a cone and plate viscometer

We investigate the flow between a shallow rotating cone and a stationary plate. This cone and plate device is used in rheometry, haemostasis as well as in food industry to study the properties of the flow w.r.t. shear stress. Physical experiments and formal computations show that close to the apex the flow is approximately azimuthal and the shear-stress is constant within the device, the quality of the approximation being controlled essentially by the single parameter Re ε2, where Re is the Reynolds number and ε the thinness of the cone-plate gap. We establish this fact by means of rigorous energy estimates and numerical simulations. Surprisingly enough, this approximation is valid though the primary flow is not itself a solution of the Navier-Stokes equations, and it does not even fulfill the correct boundary conditions, which are in this particular case discontinuous along a line, thus not allowing for a usual Leray solution. To overcome this difficulty we construct a suitable corrector

Numerical investigation of the non-isothermal contact angle

The influence of thermocapillary stress on the shape of the gas-liquid phase boundary is investigated numerically. We consider the case of a cold liquid meniscus at a heated solid wall in the absence of gravity. An "apparent contact angle" is defined geometrically and the deviation of this apparent contact angle from the prescribed static contact angle due to thermocapillary convection is studied

Computational comparison between the Taylor--Hood and the conforming Crouzeix--Raviart element

This paper is concerned with the computational performance of the P₂ P₁ Taylor-Hood element and the conforming P₂+ P-1 Crouzeix-Raviart element in the finite element discretization of the incompressible Navier-Stokes equations. To this end various kinds of discretization errors are computed as well as the behavior of two different preconditioners to solve the arising systems are studied

An operator-splitting finite-element approach to an 8:1 thermal cavity problem

[no abstract available

Numerical simulation of suspension induced rheology

summary:Flow of particles suspended in a fluid can be found in numerous industrial processes utilizing sedimentation, fluidization and lubricated transport such as food processing, catalytic processing, slurries, coating, paper manufacturing, particle injection molding and filter operation. The ability to understand rheology effects of particulate flows is elementary for the design, operation and efficiency of the underlying processes. Despite the fact that particle technology is widely used, it is still an enormous experimental challenge to determine the correct parameters for the process employed. In this paper we present 2-dimensional numerical results for the behavior of a particle based suspension and compare it with analytically results obtained for the Stokes-flow around a single particle

Finite element method for epitaxial growth with thermodynamic boundary conditions

We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height, thermodynamic boundary conditions on terrace boundaries including anisotropic line tension, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional (possibly anisotropic) "surface" diffusion of edge-adatoms along the step-edges. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. A penalty method is used in order to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements