Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

Makoto Mizuguchi; Mitsuhiro T. Nakao; Kouta Sekine; Shin’ichi Oishi

doi:10.1007/s10915-021-01636-3

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

Makoto Mizuguchi^*, Mitsuhiro T. Nakao, Kouta Sekine, Shin’ichi Oishi

^*Corresponding author for this work

School of Fundamental Science and Engineering

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

1 Citation (Scopus)

Abstract

In this paper, we propose L2(J;H01(Ω)) and L²(J; L²(Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

Original language	English
Article number	34
Journal	Journal of Scientific Computing
Volume	89
Issue number	2
DOIs	https://doi.org/10.1007/s10915-021-01636-3
Publication status	Published - 2021 Nov

Keywords

A priori error estimate
Best possible
Error constant
semi-discrete Galerkin approximation

ASJC Scopus subject areas

Software
Theoretical Computer Science
Numerical Analysis
Engineering(all)
Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

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10.1007/s10915-021-01636-3

Cite this

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title = "Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation",

abstract = "In this paper, we propose L2(J;H01(Ω)) and L2(J; L2(Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.",

keywords = "A priori error estimate, Best possible, Error constant, semi-discrete Galerkin approximation",

author = "Makoto Mizuguchi and Nakao, {Mitsuhiro T.} and Kouta Sekine and Shin{\textquoteright}ichi Oishi",

note = "Funding Information: This work was supported by CREST, JST Grant No. JPMJCR14D4, JSPS KAKENHI No.18K13462, and JSPS KAKENHI No.18K03434. Publisher Copyright: {\textcopyright} 2021, The Author(s).",

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doi = "10.1007/s10915-021-01636-3",

language = "English",

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AU - Mizuguchi, Makoto

AU - Nakao, Mitsuhiro T.

AU - Sekine, Kouta

AU - Oishi, Shin’ichi

PY - 2021/11

Y1 - 2021/11

N2 - In this paper, we propose L2(J;H01(Ω)) and L2(J; L2(Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

AB - In this paper, we propose L2(J;H01(Ω)) and L2(J; L2(Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

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KW - Best possible

KW - Error constant

KW - semi-discrete Galerkin approximation

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Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

Abstract

Keywords

ASJC Scopus subject areas

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