Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Mitsuhiro T. Nakao

doi:10.1007/BF01389881

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Mitsuhiro T. Nakao^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

28 Citations (Scopus)

Abstract

A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal order L^p error estimates are derived for 2≦p≦∞. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.

Original language	English
Pages (from-to)	139-157
Number of pages	19
Journal	Numerische Mathematik
Volume	47
Issue number	1
DOIs	https://doi.org/10.1007/BF01389881
Publication status	Published - 1985 Mar
Externally published	Yes

Keywords

Subject Classifications: AMS(MOS): 65N30, CR: G1.8

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics
Mathematics(all)

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10.1007/BF01389881

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N2 - A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal order Lp error estimates are derived for 2≦p≦∞. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.

AB - A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal order Lp error estimates are derived for 2≦p≦∞. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.

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Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Abstract

Keywords

ASJC Scopus subject areas

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