scholarly journals Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

2018 ◽  
Vol 63 (5) ◽  
pp. 503-521 ◽  
Author(s):  
Pernilla Johnsen ◽  
Tatiana Lobkova
Keyword(s):  
Parabolic Problem ◽  
Linear Parabolic Problem

scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

scholarly journals A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Liselott Flodén ◽  
Anders Holmbom ◽  
Marianne Olsson Lindberg ◽  
Jens Persson
Keyword(s):  
Spatial Scale ◽  
Parabolic Problem ◽  
Spatial Scales ◽  
Time Derivative ◽  
Local Problem ◽  
Linear Parabolic Problem ◽  
Spatial Oscillations ◽  
Spatial Oscillation

We consider the homogenization of the linear parabolic problem which exhibits a mismatch between the spatial scales in the sense that the coefficient of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

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