Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations

Irene Kyza; Charalambos Makridakis

doi:10.1137/100796819

Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations

Irene Kyza^*, Charalambos Makridakis

^*Corresponding author for this work

Applied Mathematics

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

Abstract

We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank-Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time.

Original language	English
Pages (from-to)	405-426
Number of pages	22
Journal	Siam journal on numerical analysis
Volume	49
Issue number	1
DOIs	https://doi.org/10.1137/100796819
Publication status	Published - 2011

Keywords

Backward Euler method
Blow-up solutions and rate
Conditional a posteriori estimates
Crank-Nicolson method
Duhamel's principle
Energy techniques
Fixed point arguments
Reconstruction technique
Semilinear parabolic equations

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10.1137/100796819

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abstract = "We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank-Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time.",

keywords = "Backward Euler method, Blow-up solutions and rate, Conditional a posteriori estimates, Crank-Nicolson method, Duhamel's principle, Energy techniques, Fixed point arguments, Reconstruction technique, Semilinear parabolic equations",

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AU - Kyza, Irene

AU - Makridakis, Charalambos

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AB - We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank-Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time.

KW - Backward Euler method

KW - Blow-up solutions and rate

KW - Conditional a posteriori estimates

KW - Crank-Nicolson method

KW - Duhamel's principle

KW - Energy techniques

KW - Fixed point arguments

KW - Reconstruction technique

KW - Semilinear parabolic equations

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U2 - 10.1137/100796819

DO - 10.1137/100796819

M3 - Article

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VL - 49

SP - 405

EP - 426

JO - Siam journal on numerical analysis

JF - Siam journal on numerical analysis

IS - 1

ER -

Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations

Abstract

Keywords

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