TY - JOUR
T1 - Convergence and optimality of the adaptive nonconforming linear element method for the stokes problem
AU - Hu, Jun
AU - Xu, Jinchao
N1 - Funding Information:
Acknowledgements The first author was supported by NSFC 10971005, and in part by NSFC 11031006. The second author was supported in part, by NSFC-10528102, NSF DMS 0915153, and DMS 0749202, and by the PSU-PKU Joint Center for Computational Mathematics and Applications.
PY - 2013/4
Y1 - 2013/4
N2 - 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.
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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U2 - 10.1007/s10915-012-9625-4
DO - 10.1007/s10915-012-9625-4
M3 - Article
AN - SCOPUS:84875215673
SN - 0885-7474
VL - 55
SP - 125
EP - 148
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 1
ER -