Daily Briefing: War in Ukraine: Russia Says It Is Willing to Extend Grain Deal With Ukraine

For weeks, Russia has thrown tens of thousands of soldiers into battles up and down eastern Ukraine in an attempt to drive the Ukrainians out of well-fortified positions. The deadliest fighting has been in and around Bakhmut, where both sides are trying to degrade the other in a monthslong battle of attrition, but the Ukrainian military’s General Staff is routinely reporting more than 100 attempts by Russian forces to break through their defensive lines each day.

Many of the individual battles relate as much to key roads and supply lines as to control over now devastated towns and villages. The length of the battle for Avdiivka illustrates how difficult it has been for Russian forces to take further ground in the east, where both armies have established significant defenses and Ukraine has reinforced its positions.

Russian troops fired two shells at an abandoned school in the town on Monday, killing a local woman, according to a post by Andriy Yermak, head of Ukraine’s president’s office, on the social messaging app Telegram. He posted two photographs of a three-story building that had been reduced to rubble. Russian forces also fired three other artillery rounds at the town, according to the head of the regional Ukrainian military administration, Pavlo Kyrylenko, writing on Telegram.

There was no independent confirmation of the reports. The town had a population of 31,000 before the war, but almost all its inhabitants have fled. Mr. Kyrylenko said that Russia also shelled nearby communities.

Mr. Kyrylenko said Russia’s attacks included the use of cluster munitions, which rights groups say can disproportionately harm civilians. His account could not be independently confirmed.

The Ukrainian General Staff said in a daily report that its forces had repelled attacks in the town itself as well as in at least five nearby settlements. The attacks form a pattern of destruction, according to other officials.

Russia “has been massively hitting the villages near and on the way to the town over the past week,” Vitaliy Barabash, head of Avdiivka’s military administration, said on Ukrainian television over the weekend.

“These villages are being erased,” Mr. Barabash said. He said Russian forces appeared to be “trying to cut off Avdiivka, disrupt the logistics, the delivery of personnel and ammunition,” probably in the belief that the town housed “a lot of personnel, our guys.” He added that Russian fighters had targeted a road that runs from a big industrial plant into the town.

Videos and photographs from Avdiivka posted on social media in recent weeks showed a town devastated by shelling, even by the standards of other hard-hit places in a region where fighting has raged since Russia launched its full-scale invasion of Ukraine a year ago. In one, a fire burned in a deserted industrial building, a large crater stood in a street full of debris and dozens of houses and many apartment blocks had windows blown out. Even trees appeared blackened. The material had not been confirmed independently.

Pro-Russian separatists occupied Avdiivka for a few months in 2014, when Moscow illegally annexed Crimea and set up breakaway republics in Donetsk and Luhansk, the two provinces that make up Donbas. Ukrainian forces drove the separatists out of Avdiivka in July of that year.

Over the weekend, The Institute for the Study of War, a Washington-based research group, reported a Russian claim that Moscow’s forces had taken Sjeverne and Kamianka, settlements east and west of Avdiivka. The Ukrainian military denied the claims, and there was no independent confirmation.