Discontinuous Galerkin finite element discretization for steady stokes flows with threshold slip boundary condition

This work is concerned with the discontinuous Galerkin nite approxima- tions for the steady Stokes equations driven by slip boundary condition of \friction" type. Assuming that the ow region is a bounded, convex domain with a regular boundary, we formulate the problem and its discontinuous Galerkin approximations as mixed variational inequalities of the second kind with primitive variables. The well posedness of the formulated problems are established by means of a generalization of the Babu ska-Brezzi theory for mixed problems. Finally, a priori error estimates using energy norm for both the velocity and pressure are obtained.http://www.tandfonline.com/loi/tqma20hb201

Convergence analysis of the nonconforming finite element discretization of Stokes and Navier-Stokes equations with nonlinear slip boundary conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier-Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix-Raviart element of order one, while the discrete pressure is constant element wise the inequality of Korn doe not hold. Hence we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska-Brezzi’s theory for mixed problems.http://www.tandfonline.com/loi/lnfa202018-04-10hb2017Mathematics and Applied Mathematic

A priori error analysis for Navier Stokes equations with slip boundary conditions of friction type

The time dependent Navier Stokes equations under nonlinear slip boundary conditions are discretized by backward Euler scheme in time and finite elements in space. We derive error estimates for the semi-discrete problems. The focus on the semi discrete problem in time is to obtain convergence rate without extra regularity on the weak solution by following Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). The semi discrete problem in space is analyzed with the help of the Stokes operator introduced. Finally we use the triangle inequality to derive the global a priori error estimates.The National Research Foundation of South Africahttp://link.springer.com/journal/212020-03-01hj2019Mathematics and Applied Mathematic

Finite element analysis for Stokes and Navier-Stokes equations driven by threshold slip boundary conditions

This paper is devoted to the study of finite element approximations of variational inequalities with a special nonlinearity coming from boundary conditions. After re-writing the problems in the form of variational inequalities, a fixed point strategy is used to show existence of solutions. Next we prove that the finite element approximations for the Stokes and Navier Stokes equations converge respectively to the solutions of each continuous problems. Finally, Uzawa’s algorithm is formulated and convergence of the procedure is shown, and numerical validation test is achieved.http://www.math.ualberta.ca/ijnam/am201

Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

In this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska– Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed varia-tional problem is presented and analyzed in details. Finally, numerical simulations that validate the theoret-ical findings are exhibited.http://www.degruyter.com/view/j/jnmahb2016Mathematics and Applied Mathematic

Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions

In this article, we discuss the numerical solution of the Stokes and Navier-Stokes equations completed by nonlinear slip boundary condi- tions of friction type in two and three dimensions. To solve the Stokes system, we rst reduce the related variational inequality into a saddle point-point problem for a well chosen augmented Lagrangian. To solve this saddle point problem we suggest an alternating direction method of multiplier together with nite element approximations. The solution of the Navier Stokes system combines nite element approximations, time discretization by operator splitting and augmented Lagrangian method. Numerical experiment results for two and three dimensional ow con rm the interest of these approaches.National Research Foundation of South Africa, project 85796,N00401.http://www.elsevier.com/locate/cma2017-03-31hb2016Mathematics and Applied Mathematic

Analysis of the Brinkman-Forchheimer equations with slip boundary conditions

In this work, we study the Brinkman-Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. Next we discuss the continuity of the solution with respect to Brinkman's and Forchheimer's coefficients. Finally, we show that the weak solution of the corresponding stationary problem is stable

Analysis of a time implicit scheme for the Oseen model driven by nonlinear slip boundary conditions

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtainhttp://link.springer.com/journal/212017-12-31hb2016Mathematics and Applied Mathematic

Existence results for a polymer melt with an evolving natural configuration

We consider a set of equations governing the behaviour of a polymer melt which is modelled as a viscoelastic fluid possessing a natural, or stress-free state. The natural configuration is characterized through a symmetric, proper orthogonal intermediate deformation tensor, analogous to the left Cauchy-Green deformation tensor in continuum mechanics. This tensor is required to satisfy an evolution equation. It is shown that the constraint that the intermediate tensor be proper orthogonal is satisfied provided that its initial value satisfies this constraint. Local; existence and uniqueness of solutions to the initial boundary value problem of the resulting viscoelastic fluid system are established. It is also shown that the local solutions can be extended globally provided that the data are small enough.http://www.worldscinet.com/m3as/nf201

Traveling wave solution of the Kuramoto-Sivashinsky equation : a computational study

This work considers the numerical solution of the Kuramoto-Sivashinsky equation using the fractional time splitting method. We will investigate the numerical behavior of two categories of the traveling wave solutions documented in the literature (Hooper & Grimshaw (1998)), namely: the regular shocks and the oscillatory shocks. We will also illustrate the ability of the scheme to produce convergent chaotic solutions.http://proceedings.aip.org/hb201