Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.     

A multidomain spectral collocation method for the Stokes problem

Summary We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.Deceased     

Finite element method for two‐phase immiscible flow

An explicit finite element method for numerically solving the two‐phase, immiscible, incompressible flow in a porous medium in two space dimensions is analyzed. The method is based on the use of a mixed finite element method for the approximation of the velocity and pressure a discontinuous upwinding finite element method for the approximation of the saturation. The mixed method gives an approximate velocity field in the precise form needed by the discontinuous method, which is trivially conservative and fully parallelizable in computation. It is proven that it converges to the exact solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 407–416, 1999     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

A discontinuous mixed covolume method for elliptic problems

We develop a discontinuous mixed covolume method for elliptic problems on triangular meshes. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-order L²-error estimates are derived for the approximations of both velocity and pressure.     

A mixed boundary-value problem for the wave equation in a stratified medium for high-frequency oscillations

A boundary-value problem for the wave equation in a stratified medium with mixed boundary conditions on the boundary in the case of high oscillation frequencies is considered. The Helmholtz equation for a velocity function increasing monotonically with depth is investigated. The problem is reduced to an integral equation in the high-frequency approximation, and an explicitly smooth term of its asymptotic solution is constructed.     

The Mortar Element Method with Lagrange Multipliers for Stokes Problem

In this paper,we propose a mortar element method with Lagrange multiplier for incompressible Stokes problem,i.e.,the matching constraints of velocity on mortar edges are expressed in terms of Lagrange multipliers.We also present P_1 noncon- forming element attached to the subdomains.By proving inf-sup condition,we derive optimal error estimates for velocity and pressure.Moreover,we obtain satisfactory approximation for normal derivatives of the velocity across the interfaces.     

Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L². Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016     

A Second Order Nonconforming Triangular Mixed Finite Element Scheme for the Stationary Navier-Stokes Equations

In this paper,a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as approximation space for the velocity and the linear element for the pressure.The convergence analysis is presented and optimal error estimates of both broken H1-norm and L2-norm for velocity as well as the L2-norm for the pressure are derived.     

Haar wavelet approximation for magnetohydrodynamic flow equations

This study proposes Haar wavelet (HW) approximation method for solving magnetohydrodynamic flow equations in a rectangular duct in presence of transverse external oblique magnetic field. The method is based on approximating the truncated double Haar wavelets series. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The calculations show that the accuracy of the Haar wavelet solutions is quite good even in the case of a small number of grid points. The HW approximation method may be used in a wide variety of high-order linear partial differential equations. Application of the HW approximation method showed that it is reliable, simple, fast, least computation at costs and flexible.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.     

A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows

This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interior of elements, respectively, and piecewise polynomials of degrees k and k + 1 for the boundary parts of the velocity and pressure, respectively. Wellposedness of the discrete scheme is established. The method yields a globally divergence-free velocity approximation. Optimal priori error estimates are derived for the velocity gradient and pressure approximations. Numerical results are provided to confirm the theoretical results.     

Approximation Results for Kinetic Variants of TSP

   Abstract. We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results: • If the points all move with the same velocity, then there is a polynomial time approximation scheme for the Kinetic TSP. • The Kinetic TSP cannot be approximated better than by a factor of 2 by a polynomial time algorithm unless P = NP, even if there are only two moving points in the instance. • The Kinetic TSP cannot be approximated better than by a factor of

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k (k ≥ 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.     

Superconvergence of a combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem

A combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is studied. That is to say, the mixed finite element method is applied to the flow equation, and the transport equation is solved by an interior penalty discontinuous Galerkin method. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is obtained. Numerical experimental results are presented to verify the theoretical analysis.     

Track method for the calculation of plasma heating by charged thermonuclear reaction products for axisymmetric flows

Integral formulas for the three-dimensional case that give the plasma heating rate per unit volume are obtained using the track method and by integrating the well-known Cauchy problem for the steady-state homogeneous kinetic equation in the Fokker–Planck approximation in the absence of diffusion of the distribution function in the velocity space and under the condition that the velocity of the produced particles is independent on the direction of their escape. It is shown that both integral formulas are equivalent and, in the case of space homogeneous coefficients, turn into the model of local plasma heating away from the domain boundary. In addition to the known direct track method, the inverse method based on the approximation of the integral formula is developed. It is shown that the accuracy of the direct method is significantly decreased in the vicinity of the symmetry axis for not very fine angular grids. In the inverse method, the accuracy is not lost. It is shown that the computational cost of the inverse method can be significantly reduced without the considerable reduction of the computation accuracy.     

Error estimates for an operator-splitting method for Navier-Stokes equations: Second-order schemes

We present in this paper an error analysis of a fractional-step method for the approximation of the unsteady incompressible Navier-Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain second-order error estimates in the time step size, both for velocity and pressure. Numerical results in agreement with the error analysis are also presented.     

Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet

This paper deals with the unsteady axisymmetric flow and heat transfer of a viscous fluid over a radially stretching sheet. The heat is prescribed at the surface. The modelled non-linear partial differential equations are solved using an analytic approach namely the homotopy analysis method. Unlike perturbation technique, this approach gives accurate analytic approximation uniformly valid for all dimensionless time. The explicit expressions for velocity, temperature and skin friction coefficient are developed. The influence of time on the velocity, temperature and skin friction coefficient is discussed.     

Analytical Formulation of the Classical Friction Model for Motion Analysis and Simulation

Starting from a precise definition of friction torque when velocity vanishes that distinguishes the case of instantaneous zero crossing from that where the velocity is zero over a time interval, this paper proposes a compact analytical formulation of the classical discontinuous friction model that is useful for motion analysis. A finite state machine that allows a numerically robust computation of motion equations when velocity vanishes or motion restarts is then defined. Simulation results show that the discontinuous model can be seen as an asymptotic approximation, infinitely fast, of a recently proposed continuous, dynamic friction model.