Constructive a priori error estimates for a full discrete approximation of the heat equation

Mitsuhiro T. Nakao, Takuma Kimura, Takehiko Kinoshita

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations. This implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems. Our results can also be considered as an explicit optimal estimate with the limited regularity of solutions, which should be unknown up to the present.

Original languageEnglish
Pages (from-to)1525-1541
Number of pages17
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number3
DOIs
Publication statusPublished - 2013
Externally publishedYes

Fingerprint

A Priori Error Estimates
Discrete Approximation
Heat Equation
Numerical Verification
Semidiscretization
Galerkin Finite Element Method
Regularity of Solutions
Nonlinear Parabolic Equations
A Posteriori Error Estimates
Galerkin methods
Fundamental Solution
Estimate
Nonlinear Problem
Interpolation
Exact Solution
Interpolate
Numerical Solution
Imply
Unknown
Hot Temperature

Keywords

  • Constructive a priori error estimates
  • Galerkin methods
  • Parabolic problem

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

Constructive a priori error estimates for a full discrete approximation of the heat equation. / Nakao, Mitsuhiro T.; Kimura, Takuma; Kinoshita, Takehiko.

In: SIAM Journal on Numerical Analysis, Vol. 51, No. 3, 2013, p. 1525-1541.

Research output: Contribution to journalArticle

Nakao, Mitsuhiro T. ; Kimura, Takuma ; Kinoshita, Takehiko. / Constructive a priori error estimates for a full discrete approximation of the heat equation. In: SIAM Journal on Numerical Analysis. 2013 ; Vol. 51, No. 3. pp. 1525-1541.
@article{36583e47c82a4e5c8d442fbb6232c963,
title = "Constructive a priori error estimates for a full discrete approximation of the heat equation",
abstract = "In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations. This implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems. Our results can also be considered as an explicit optimal estimate with the limited regularity of solutions, which should be unknown up to the present.",
keywords = "Constructive a priori error estimates, Galerkin methods, Parabolic problem",
author = "Nakao, {Mitsuhiro T.} and Takuma Kimura and Takehiko Kinoshita",
year = "2013",
doi = "10.1137/120875661",
language = "English",
volume = "51",
pages = "1525--1541",
journal = "SIAM Journal on Numerical Analysis",
issn = "0036-1429",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Constructive a priori error estimates for a full discrete approximation of the heat equation

AU - Nakao, Mitsuhiro T.

AU - Kimura, Takuma

AU - Kinoshita, Takehiko

PY - 2013

Y1 - 2013

N2 - In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations. This implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems. Our results can also be considered as an explicit optimal estimate with the limited regularity of solutions, which should be unknown up to the present.

AB - In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations. This implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems. Our results can also be considered as an explicit optimal estimate with the limited regularity of solutions, which should be unknown up to the present.

KW - Constructive a priori error estimates

KW - Galerkin methods

KW - Parabolic problem

UR - http://www.scopus.com/inward/record.url?scp=84884991820&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884991820&partnerID=8YFLogxK

U2 - 10.1137/120875661

DO - 10.1137/120875661

M3 - Article

VL - 51

SP - 1525

EP - 1541

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 3

ER -