Optimal error bound of restricted Monte Carlo integration on anisotropic Sobolev classes*

2006 ◽  
Vol 16 (6) ◽  
pp. 588-593 ◽  
Author(s):  
Gao Wenhua ◽  
Ye Peixin ◽  
Wang Hui
Keyword(s):  
Monte Carlo ◽  
Error Bound ◽  
Monte Carlo Integration ◽  
Sobolev Classes ◽  
Optimal Error ◽  
Optimal Error Bound

OPTIMAL ERROR BOUND AND APPROXIMATION METHODS FOR A CAUCHY PROBLEM OF THE MODIFIED HELMHOLTZ EQUATION

2011 ◽  
Vol 09 (02) ◽  
pp. 305-315 ◽  
Author(s):  
AILIN QIAN ◽  
YUJIANG WU
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Error Bound ◽  
Optimality Theory ◽  
Tikhonov Regularization Method ◽  
Regularization Methods ◽  
Optimal Error ◽  
Modified Helmholtz Equation ◽  
Optimal Error Bound

We consider a Cauchy problem for a modified Helmholtz equation, especially when we give the optimal error bound for this problem. Some spectral regularization methods and a revised Tikhonov regularization method are used to stabilize the problem from the viewpoint of general regularization theory. Hölder-type stability error estimates are provided for these regularization methods. According to the optimality theory of regularization, the error estimates are order optimal.

scholarly journals An embedded discontinuous Galerkin method for the Oseen equations

2021 ◽  
Author(s):  
Yongbin Han ◽  
Yanren Hou
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Error Bound ◽  
Discontinuous Galerkin Method ◽  
Convergence Order ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Oseen Equations ◽  
Optimal Error Bound ◽  
Convection Dominated

In this paper, the a prior error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in the L 2 (Ω) norm, has an optimal error bound with convergence order k + 1, where the constants are dependent on the Reynolds number (or ν − 1 ), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence order k +1 / 2. Here, k is the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results.

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