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Maximum Norm (maximum + norm)

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Engineering60%

Selected Abstracts

Some superconvergence results for the covolume method for elliptic problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2001
Jianguo Huang
Abstract In this paper, we attempt to give analysis of the covolume method for solving general self-adjoint elliptic problems. We first present some useful superconvergence results for the deviation between the solution of the covolume method and the solution of the induced finite element method, in the energy norm and maximum norm, respectively. With these results, we then reproduce the maximum norm estimates obtained by Chou and Li for the covolume method easily. Furthermore, based on the covolume method, we propose a high-accuracy algorithm for solving general self-adjoint elliptic problems. Compared with the original covolume method, the computation work of the new algorithm is increased slightly, but the approximate error is improved remarkably. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010
Hong-Lin Liao
Abstract Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth-order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher-order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source]

Compact difference schemes for heat equation with Neumann boundary conditions

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009
Zhi-Zhong Sun
Abstract In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949,959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(,2 + h2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(,2 + h3). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600,616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]

On Existence Theorems for Solutions of Non-Linear Systems

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Jan Mayer
An equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton-Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G , ,n, G , ,n, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton-Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized. [source]

The streamline,diffusion method for a convection,diffusion problem with a point source

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Görg Roos Prof. Dr.
A linear singularly perturbed convection,diffusion problem with a point source is considered. The problem is solved using the streamline,diffusion finite element method on a class of Shishkin,type meshes. We prove that the method is almost optimal with uniform second order of convergence in the maximum norm. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results. [source]