303 research outputs found
Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems
In this paper we develop a family of preconditioners for the linear algebraic
systems arising from the arbitrary Lagrangian-Eulerian discretization of some
fluid-structure interaction models. After the time discretization, we formulate
the fluid-structure interaction equations as saddle point problems and prove
the uniform well-posedness. Then we discretize the space dimension by finite
element methods and prove their uniform well-posedness by two different
approaches under appropriate assumptions. The uniform well-posedness makes it
possible to design robust preconditioners for the discretized fluid-structure
interaction systems. Numerical examples are presented to show the robustness
and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2.
Rerun all the numerical tests 3. changed title, abstract and corrected lots
of typos and inconsistencies 4. added reference
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
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