As the fighting raged, the Kremlin announced that Russian President Vladimir Putin would meet his Ukrainian counterpart Petro Poroshenko on August 26 in Minsk, Belarus. The two leaders have not met since early June, despite a rapidly climbing death toll in east Ukraine.
One soldier was killed and four wounded Tuesday when a volunteer battalion came under mortar fire before entering the town of Ilovaysk, 18 kilometers (11 miles) east of Donetsk, interior minister Arsen Avakov said on Facebook.
Among the wounded was the commander of the Donbass battalion, Semyon Semenchenko, who said government soldiers had destroyed three rebel checkpoints and four firing positions and that fighting continued.
Government efforts to quell the pro-Russian separatists have focused on encircling Donetsk, the largest rebel-controlled city in eastern Ukraine. Fighting began in mid-April after Russia annexed the southern Ukrainian peninsula of Crimea but in the last few weeks, the government has recaptured significant amounts of rebel territory.
Ukrainian troops were also advancing in the separatist region of Luhansk, capturing one neighborhood in the city of Luhansk as they battled the rebels Tuesday on city streets, Col. Andriy Lysenko, a spokesperson for Ukraine's National Security Council, told reporters.
The fighting in eastern Ukraine has forced nearly 344,000 people to flee their homes, according to the United Nations - a number that has grown in recent weeks as living conditions in rebel-held cities deteriorates rapidly.
With the rebels losing more and more ground, the Kremlin announced the meeting in Minsk, which would also include officials from the European Commission and the Eurasian Customs Union, which is comprised of Russia, Kazakhstan, and Belarus.
Poroshenko, who confirmed the meeting, said "stabilizing the situation" in eastern Ukraine would be a key topic of discussion. Putin spokesman Dmitry Peskov said the Russian leader wanted to talk about the deteriorating humanitarian situation there.
Valery Chaly, deputy head of the Ukrainian presidential administration, said in a televised briefing that the government is seeing a "clear diplomatic roadmap" ahead of them and expressed hope that they will be able to come up with new approaches to end the war.
Living standards appear to be particularly dire in Luhansk, near the Russian border, which has been left without electricity, running water or phone connections for 17 days.
Luhansk city hall said the city center came under fierce shelling overnight, which killed and wounded an unspecified number of civilians. Residents are standing in lines to buy bread as food supplies are running out, it said. Authorities also raised the alarm about a potential outbreak of infectious diseases since household garbage has not been taken away for more than two weeks.
In Donetsk, artillery fire was heard across town, the city hall said in a statement. But the city's suburbs seem to be the most hard-hit.
A resident of Olenivka, a village south of Donetsk, told the Associated Press by telephone that Ukrainian government troops were firing artillery from fields near her house.
Tatyana, who asked for her surname not to be printed for fear of reprisal from Ukrainian troops, said that houses in her village had been hit by return fire in previous days. Loud blasts could be heard in the background as she spoke.
The shelling appeared to be aimed at Donetsk's Petrovsky district, which has come under sustained rocket attack over the past week, she said.
Several residents in the Petrovsky district have told the Associated Press that rebels have positioned and fired Grad multiple missile launchers in their neighborhood, which appears to have drawn the incoming fire.
The eyewitness accounts appear to fly in the face of claims by the Kiev government that the armed forces have refrained from aiming rockets at residential areas.
The government, meanwhile, has accused the rebels of killing dozens of civilians in a shelling attack Monday on a convoy of refugees fleeing Luhansk. The rebels denied that any attack took place, while the US confirmed the shelling of the convoy but said it did not know who was responsible.
There have been no eyewitness accounts of what happened and the war-torn area is largely off limits for journalists, making independent verification impossible.
The Ukrainian defence ministry released a video Tuesday that reportedly showed people who survived that attack at the nearby village of Novosvitlovka. The video showed a young man lying on a hospital bed sobbing. A woman who was not pictured said the man's mother was killed in the attack. The video could not be independently verified.
Journalists still have not seen any video or photos of the scene of Monday's shelling attack on the road from Luhansk to Russia.
German Chancellor Angela Merkel will travel to Kiev for the first time since the crisis erupted on Saturday to meet with top officials.
Ukraine and the West have voiced concerns about Russia's military activity near the Ukrainian border. Moscow has said it can do what it wants on its territory but invited a mission from the Organization for Security and Cooperation in Europe to observe two border crossings in a bid to counter claims that Russia is supplying the rebels with weapons.
Paul Picard, head of that OSCE mission, told reporters in a Russian border town on Tuesday that observers had seen a marked increase of military activity around the border points over the past week, including Russian helicopter activity. But he said no helicopters were observed crossing the border.
Picard also said the observers had seen "groups of young men and women wearing military-style dress" moving back and forth across the border. No weapons or military vehicles were observed.
Tensions have been high over the past week since Russia said it plans to send a massive aid convoy of over 200 trucks to help people in eastern Ukraine. The International Committee of the Red Cross, which is expected to take responsibility for the convoy when it enters eastern Ukraine, was still waiting for security guarantees from all sides Tuesday.
The Russian foreign ministry said foreign minister Sergey Lavrov and ICRC president Peter Maurer agreed to take steps "in the next several hours" to clear the way for the convoy to proceed to Ukraine.
The Russian foreign ministry said both Moscow and the rebels have provided security guarantees for the convoy, but not Kiev. Ukraine's foreign ministry spokesperson Evhen Perebiynis, said Tuesday that Kiev cannot guarantee the safety of the convoy on rebel territory, since it does not control that area.
Russia chose to drive the convoy close to a rebel-held border post against Ukraine's wishes.
Also on Tuesday, Malaysian defence minister Hishammuddin Hussein said the first remains of Malaysians who were killed when a jetliner was shot down over Ukraine will be flown home this week.
All 298 people on board died when Malaysia Airlines Flight 17 was shot down on July 17 over an area controlled by pro-Russia separatists.