Russia on Tuesday pressed more troops into Ukraine and continued its offensive to take control of the Donbas region in the country's east.?

Multiple cities and towns came under Russian attack, including eastern cities of Kharkiv and Kramatorsk and areas around Zaporizhzhia and Dnipro west in Donbas region.

The Russian control of Donbas region, Ukraine's industrial heartland of coal mines and factories, would mean slicing of the country into two and would serve as a badly-needed victory for President Vladimir Putin.

Russian Defense Ministry spokesman Major General Igor Konashenkov said air-launched missiles destroyed 13 Ukrainian troop and weapons locations, while the air force struck 60 other Ukrainian military facilities, including missile warhead storage depots.

Russian artillery hit nearly 1,300 Ukrainian military facilities and over 1,200 troop concentrations over the past 24 hours, Konashenkov said. The claims could not be independently verified.

In what both sides described as a new phase of the war, the Russian assault began on Monday along a boomerang-shaped front stretching more than 300 miles (480 kilometers) from northeastern Ukraine to the country's southeast. Ukraine's military said Russian forces tried to "break through our defenses along nearly the entire front line”.

Weeks ago, after the abortive Russian push to take Kyiv, the Kremlin declared that its main goal was the capture of the mostly Russian-speaking Donbas, where Moscow-backed separatists have been fighting Ukrainian forces for eight years. A Russian victory in the Donbas would deprive Ukraine of the industrial assets concentrated there, including mines, metals plants and heavy-equipment factories.

A senior US defense official, speaking on condition of anonymity to discuss the Pentagon's assessments of the war, said the Russians had added two more combat units, known as battalion tactical groups, in Ukraine over the preceding 24 hours. That brought the total number of units in the country to 78, all of them the south and the east, up from 65 last week. That would translate to about 55,000 to 62,000 troops.

A European official, likewise speaking on condition of anonymity to discuss military assessments, said Russia also has 10,000 to 20,000 foreign fighters in the Donbas. They are a mix of mercenaries from Russia's private Wagner Group and Russian proxy fighters from Syria and Libya, according to the official.

While Ukraine portrayed the attacks on Monday as the start of the long-feared offensive in the east, some observers noted that an escalation has been underway there for some time and questioned whether this was truly the start of a new offensive.

The US official said that the offensive in the Donbas has begun in a limited way, mainly in an area southwest of the city of Donetsk and south of Izyum.

Justin Crump, a former British tank commander now with the strategic advisory company Sibylline, said the Ukrainian comments could, in part, be an attempt to persuade allies to send more weapons.

“What they're trying to do by positioning this, I think, is focus people's minds and effort by saying, 'Look, the conflict has begun in the Donbas',” Crump said. “That partly puts pressure on NATO and EU suppliers to say, 'Guys, we're starting to fight now. We need this now'.”

European and American arms have been key to bolstering Ukraine's defense, helping the under-gunned country to hold off the Russians. Dutch Prime Minister Mark Rutte told Ukrainian President Volodymyr Zelenskyy on Tuesday that the Netherlands would send “heavier material", including armored vehicles.

Associated Press journalists in Kharkiv said at least four people were killed and three wounded in a Russian attack on a residential area of the city. The attack occurred as residents attempted to maintain a sense of normalcy, with municipal workers planting spring flowers in public areas.

An explosion also rocked Kramatorsk, killing at least one person and wounding three, according to AP journalists at the scene.

Eyewitness accounts and reports from officials have given a broad picture of the extent of the Russian advance. But independent reporting in the parts of the Donbas held by Russian forces and separatists is severely limited, making it difficult to know what is happening in many places on the ground.

Military experts said the Russians' goal is to encircle Ukrainian troops from the north, south and east.

Key to the campaign is the capture of Mariupol, the now-devastated city in the Donbas that the Russians have besieged since the early days of the war. Taking Mariupol would deprive Ukraine of a vital outlet to the sea and complete a land bridge between Russia and the Crimean Peninsula, seized from Ukraine from 2014.

It would also free up Russian troops to move elsewhere in the Donbas.

A few thousand Ukrainian troops, by the Russians' estimate, remained holed up in a sprawling Mariupol steel plant, representing what was believed to be the last major pocket of resistance in the city.

On Tuesday, Russia issued a new ultimatum to the Ukrainian defenders to surrender, saying those who come out will “keep their lives," and said a cease-fire was being declared in the area so the combatants could leave the plant.

The Ukrainians have ignored previous such offers, and there was no immediate confirmation a cease-fire occurred.

The Kremlin-backed leader of Chechnya, Ramzan Kadyrov, whose forces have taken part in the fighting in Mariupol, said on a messaging app that Russian forces would root out the Ukrainian resistance within hours and take full control of the steel mill on Tuesday.

Kadyrov is known for his bluster and has repeatedly predicted the city's fall in the past.

With AP inputs