Multilevel discretization of symmetric saddle point systems without the discrete LBB condition

Authors:
Constantin Bacuta;Peter Monk
Affiliations:
University of Delaware, Department of Mathematics, 501 Ewing Hall 19716, United States;University of Delaware, Department of Mathematics, 501 Ewing Hall 19716, United States
Venue:
Applied Numerical Mathematics
Year:
2012

Citing 11
Cited 1

Mixed and hybrid finite element methods

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Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates

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An Optimal Adaptive Finite Element Method for the Stokes Problem

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Schur complements on Hilbert spaces and saddle point systems

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Numerical Approximation of Partial Differential Equations

Multilevel Gradient Uzawa Algorithms for Symmetric Saddle Point Problems

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Abstract

Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddle-point problems. The discrete stability Ladyshenskaya-Babusca-Brezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel sequence of nested approximation spaces for the constrained variable. The main idea is to maintain an accurate representation of the residual associated with the main equation at each step of the inexact Uzawa algorithm at the continuous level. The residual representation is approximated by a Galerkin projection. Whenever a sufficient condition for the accuracy of the representation fails to be satisfied, the representation of the residual is projected on the next (larger) space available in the prescribed multilevel sequence. Numerical results supporting the efficiency of the algorithms are presented for the Stokes equations and a div-curl system.