Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

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

Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory

A Unified Approach for Uzawa Algorithms

International audienceWe present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite-or infinite-dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition