scholarly journals On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 716
Author(s):  
Ahlam Arama ◽  
Shuhuang Xiang ◽  
Suliman Khan
Keyword(s):  
Convergence Rate ◽  
Error Bound ◽  
Optimal Error ◽  
Numerical Examples ◽  
Jacobi Weights ◽  
Jacobi Weight ◽  
Optimal Error Bound ◽  
Theoretical Results

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results.

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

2015 ◽  
Vol 5 (4) ◽  
pp. 301-311 ◽  
Author(s):  
Lijun Yi
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Error Bound ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Time Stepping ◽  
Theoretical Results

AbstractThe h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

Functional-discrete Method (FD-method) for Matrix Sturm-Liouville Problems

2005 ◽  
Vol 5 (4) ◽  
pp. 362-386 ◽  
Author(s):  
B. Ĭ. Bandyrskiĭ ◽  
I. P. Gavrilyuk ◽  
I. I. Lazurchak ◽  
V. L. Makarov
Keyword(s):  
Asymptotic Behavior ◽  
Convergence Rate ◽  
Geometric Progression ◽  
Order Number ◽  
Discrete Method ◽  
Numerical Examples ◽  
Matrix Coefficients ◽  
Sturm Liouville ◽  
Theoretical Results

AbstractA new algorithm for Sturm|Liouville problems with matrix coefficients is proposed which possesses the convergence rate of a geometric progression with a denominator depending inversely proportional on the order number of eigenvalues. The asymptotic behavior of the distance between neighboring eigenvalues if the order number tends to infinity is investigated too. Numerical examples confirming the theoretical results are given.

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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