Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Inexact and preconditioned Uzawa algorithms for saddle point problems
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
An efficient smoother for the Stokes problem
Applied Numerical Mathematics - Special issue on multilevel methods
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
SIAM Journal on Numerical Analysis
Computational scales of Sobolev norms with application to preconditioning
Mathematics of Computation
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates
SIAM Journal on Numerical Analysis
An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the Inf-Sup Condition
SIAM Journal on Numerical Analysis
Regularity estimates for elliptic boundary value problems in Besov spaces
Mathematics of Computation
Optimal relaxation parameter for the Uzawa Method
Numerische Mathematik
An Optimal Adaptive Finite Element Method for the Stokes Problem
SIAM Journal on Numerical Analysis
Schur complements on Hilbert spaces and saddle point systems
Journal of Computational and Applied Mathematics
Numerical Approximation of Partial Differential Equations
Numerical Approximation of Partial Differential Equations
Multilevel discretization of symmetric saddle point systems without the discrete LBB condition
Applied Numerical Mathematics
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
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In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the continuous level with a gradient type algorithm at the discrete level. The algorithm is based on the existence of a priori multilevel sequences of nested approximation pairs of spaces, but the family does not have to be stable. To ensure convergence, the process has to maintain an accurate representation of the residuals at each step of the inexact Uzawa algorithm at the continuous level. The residual representations at each step are approximated by projections or representation operators. Sufficient conditions for ending the iteration on a current pair of discrete spaces are determined by computing simple indicators that involve consecutive iterations. When compared with the standard Uzawa multilevel algorithm, our proposed algorithm has the advantages of automatically selecting the relaxation parameter, lowering the number of iterations on each level, and improving on running time. By carefully choosing the discrete spaces and the projection operators, the error for the second component of the solution can be significantly improved even when comparison is made with the discretization on standard families of stable pairs.