# Convergence and optimality of the adaptive nonconforming linear element method for the stokes problem

Jun Hu, Jinchao Xu

Research output: Contribution to journalArticle

21 Citations (Scopus)

### Abstract

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.

Original language English (US) 125-148 24 Journal of Scientific Computing 55 1 https://doi.org/10.1007/s10915-012-9625-4 Published - Apr 1 2013

### Fingerprint

Stokes Problem
Error Estimator
Optimality
Prolongation
Reliability Analysis
Approximation Error
Operator
Orthogonality
Interpolate
Oscillation
Dependent
Reliability analysis
Estimate
Mathematical operators
Interpolation
Standards
Class

### All Science Journal Classification (ASJC) codes

• Software
• Theoretical Computer Science
• Numerical Analysis
• Engineering(all)
• Computational Theory and Mathematics
• Computational Mathematics
• Applied Mathematics

### Cite this

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abstract = "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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In: Journal of Scientific Computing, Vol. 55, No. 1, 01.04.2013, p. 125-148.

Research output: Contribution to journalArticle

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AB - 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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