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The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity–pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments.

The paper analyzes the Stokes system with threshold slip boundary conditions of Navier type. Based on the fixed-point formulation we prove the existence of a solution for a class of solution-dependent slip functions g satisfying an appropriate growth condition and its uniqueness provided that g is one-sided Lipschitz continuous. Further we study under which conditions the respective fixed-point mapping is contractive. To discretize the problem we use P1-bubble/P1 elements. Properties of discrete models in dependence on the discretization parameter are analysed and convergence results are established. In the second part of the paper we briefly describe the duality approach used in computations and present results of a model example.

In this work, a finite element approximation of the Stokes problem under a slip boundary condition of friction type, known as the Tresca boundary condition, is considered. We treat the approximate problem of a four field mixed formulation using the $${\mathbb {P}}^{1}$$ -bubble element for the velocity field, $${\mathbb {P}}^{1}$$ element for the pressure field and the $${\mathbb {P}}^{1}$$ element for the Lagrange multipliers $$\lambda _{n}$$ and $$\lambda _{t}$$ defined on the slip boundary. The multiplier $$\lambda _{t}$$ is introduced to regularize the non-differentiable problem, whereas $$\lambda _{n}$$ treats the impermeability condition. Existence and uniqueness results for both continuous and discrete problems are proven and an a priori error estimate is established. Numerical realization of such problem is discussed and some numerical tests are provided.

The article deals with the Stokes flow under the leak boundary conditions. The problem is discretized using the mixed finite element approximation and solved as algebraic optimization problem. The respective optimality conditions are the starting point for the algorithm based on an active set implementation of the semi-smooth Newton method. Numerical experiments compare performance of the algorithm with the path-following interior point method.

The Stokes flow with the stick-slip boundary condition combining the Navier and Tresca laws is considered. The finite element approximation leads to an algebraic optimization problem. Its optimality condition is the starting point for an active set implementation of the semi-smooth Newton method. Numerical experiments demonstrate computational efficiency of an adaptive diagonal preconditioner.

In this Note we propose and study a three field mixed formulation for solving the Stokes problem with Tresca-type nonlinear boundary conditions. Two Lagrange multipliers are used to enforce div ( u ) = 0 constraint and to regularize the energy functional. The resulting problem is discretized using P1 bubble/P1-P1 finite elements. Optimal error estimate is derived and a numerical validation test is achieved.

In this paper, the convergence of a Neumann-Dirichlet algorithm to approximate Coulomb's contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

Let F be a nonlinear mapping defined from a Hilbert space X into its dual X′, and let x be in X the solution of F(x)=0. Assume that, a priori, the zone where the gradient of the function x has a large variation is known. The aim of this article is to prove a posteriori error estimates for the problem F(x)=0 when it is approximated with a Petrov–Galerkin finite element method combined with a domain decomposition method with nonmatching grids. A residual estimator for a model semi-linear problem is proposed. We prove that this estimator is asymptotically equivalent to a simplified one adapted to parallel computing. Some numerical results are presented, showing the practical efficiency of the estimator.

In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

In this Note, we deal with the domain decomposition method with non-matching grids for solving unilateral contact problem without friction between two deformable bodies. The unilateral contact conditions at the interface are expressed with a mixed formulation. The space of Lagrange multiplier defined on the contact zone is approximated by piecewise constant functions. Optimal error bound of h3/4 is obtained.

27p, 4 fig; International audience; We observe $n$ heteroscedastic stochastic processes $\{Y_v(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t \in [0,1]$, $Y_v(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_v$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_1,\ldots,g_n$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the "BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions.

The Stokes problem plays an important role in computational fluid dynamics since it is encountered in the time discretization of (incompressible) Navier-Stokes equations by operator-splitting methods [2, 3]. Space discretization of the Stokes problem leads to large scale ill-conditioned systems. The Uzawa (preconditioned) conjugate gradient method is an efficient method for solving the Stokes problem. The Uzawa conjugate gradient method is a decomposition coordination method with coordination by a Lagrange multiplier.

In this work we propose and study a three field mixed formulation for solving the Stokes problem with Tresca-type non-linear boundary conditions. Two Lagrange multipliers are used to enforce div(u)=0 constraint and to regularize the energy functional. The resulting problem is discretised using "P1 bubble/P1-P1" finite elements. Error estimates are derived and several numerical studies are achieved.

In this paper, we propose and study a Robin domain decomposition algorithm to approximate a frictionless unilateral problem between two elastic bodies. Indeed this algorithm combines, in the contact zone, the Dirichlet and Neumann boundaries conditions (Robin boundary condition). The primary feature of this algorithm is the resolution on each sub-domain of variational inequality.

In this article, we propose and study different mixed variational methods in order to approximate the Signorini problem with friction using finite elements. The discretized normal and tangential constraints at the contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle−point formulation. A priori error estimates are established and several numerical examples corresponding to the different choices of the discretized normal and tangential constraints are carried out. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006