Hi Robert,
after my machine crashed I rebooted it. And used to import the ascii 
files in the way that you proposed again and now it is working fine! So 
I think the very long calculation time must have been caused by my OS. 
If I do the import of the ascii files like this:
/
library(raster)

modeled_years=30
x1 <- modeled_years+1
#Month01
#list ascii files
asc_files <- list.files(pattern = glob2rx('EVAPTR_dem30m_*01.sum'))

#unlist_files like "EVAPTR_dem30m_Layer1_*01.sum"
asc_files <- asc_files[-c(x1:length(asc_files))]

#asc_files is now a list of 30 files 1971-2000 for the month January
asc_files
# with multiple files, use stack rather than brick
EtRR01 <- stack(asc_files)
EtRR01.sum <- calc(EtRR01, mean, overwrite=TRUE)

#Month02
asc_files <- list.files(pattern = glob2rx('EVAPTR_dem30m_*02.sum'))
asc_files<- asc_files[-c(x1:length(asc_files))]
asc_files
# with multiple files, use stack rather than brick
EtRR02 <- stack(asc_files)
EtRR02.sum <- calc(EtRR02, mean, overwrite=TRUE)

#Month03
...
/
R is doing the import of all 360 files and the calculation of the 
temporal mean for every month of the year within 2 minutes!

Thank you very much for your comments!
Best,
Swen




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