The city, which had a pre-conflict population of 900,000, rocked to the crash of shells and gunfire over the weekend and heavy guns boomed through the night into Monday from the outskirts of the city.
There was no definitive word on casualties from either side from the weekend assault by the government.
One artillery shell hit a high-security prison on the city's western outskirts late on Sunday, killing one inmate and injuring three others, the city council said. More than 100 inmates escaped from the "strict regime" prison for dangerous criminals after the shell struck - though some returned later.
Further shelling on Monday from the direction of the international airport and Yasynuvata to the north knocked out a string of power stations, the municipal authority added.
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Government forces called on the rebels to surrender on Sunday. The separatists, who have proclaimed "people's republics" in the Russian-speaking east, have said there will be no end to fighting until Kiev withdraws its troops.
'Tightening the ring'
Though the government says it is tightening a cordon around the separatists in Donetsk amid changes in their leadership and desertions in their ranks, swathes of the east are still under rebel control including the big border city of Luhansk, Horlivka to the north of Donetsk and Makiyivka to its east.
U.N. agencies say more than 1,100 people have been killed in four months of fighting between the separatists who seek union with Russia and troops representing a pro-Western government.
Though Lysenko said government forces had cut off the Donetsk-based rebels from their comrades in Luhansk, the key town of Krasny Luch, which lies between the two cities, is still not under government control, military sources in Kiev said.
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Krasny Luch is a rail and road junction through which Russian military equipment has been transported to the rebels, Kiev says.
Ukraine and its Western allies accuse Russia of fomenting the separatist revolt which erupted in April after Russia's annexation of Ukraine's Crimea peninsula.
It says Russia is funnelling tanks and missile systems to the rebels. Moscow denies involvement.
Donetsk, a once bustling metropolis, is facing an increasing shortage of food, water and fuel. Few people are on the streets, though groups of armed separatist fighters can be seen. There is relatively little traffic, with petrol in short supply.
Those who have not left for the countryside are staying indoors. Banks are closed and pensions and social allowances are not being paid.
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Though all sides recognise the seriousness of the humanitarian situation in eastern Ukraine, Kiev and its Western allies suspect Russia could use the situation to move its forces into the country.
Kiev said on Saturday it had headed off, by diplomatic means, an attempt by Russia to send troops into Ukraine under the guise of peacekeepers accompanying a humanitarian convoy sanctioned by the Red Cross. Moscow dismissed the allegation as a "fairy tale".
The geopolitical tussle over the future of the ex-Soviet state of 46 million has grown sharper since the July 17 downing of a Malaysian airliner in the eastern Ukraine conflict zone, with the deaths of all 298 passengers and crew.
Kiev and its Western allies have laid the blame for the attack at the door of the pro-Russian rebels. The separatists and Moscow say flight MH17 was downed as a result of Ukraine's military offensive.
- By Reuters