Residents of separatist-controlled regions in eastern Ukraine voted Sunday to elect legislators and executives in polls that have been staunchly denounced by the international community.

Voting in the main rebel city of Donetsk proceeded in the presence of gunmen inside three polling stations visited by the AP.

Despite a cease-fire agreement being signed in September, fighting continues almost daily between government troops and rebel forces in the area.

Election organizers have cast the vote as a decisive development in the break from Ukrainian rule by the mainly Russian-speaking regions of Donetsk and Luhansk. Around half the territory of those areas is held by separatist forces.

Russia supports the elections, but the UN, EU and the United States say it violates Ukraine's constitution and the terms of the cease-fire. The truce deal, which has been signed by rebel leaders, Ukrainian and Russian officials, envisions local elections being held across the whole of the east, but under Ukrainian law.

Alexander Zakharchenko, whose election as head of the self-styled Donetsk People's Republic is a foregone conclusion, said Sunday that he hoped the vote would bring peace to a region where 4,000 people have been killed in fighting.

"If they give us recognition and return the land we've lost without putting up a fight, then we will restore normal economic ties (with Ukraine) and we will live like equal economic partners," said Zakharchenko, who has been leading the rebel government since early August.

The eve of the vote saw an intensification of hostilities around Donetsk, but there was little sign of unrest Sunday.

It is unclear exactly how many people were able to vote as rebel officials say they have no access to central Ukrainian electoral rolls. Roman Lyagin, head of the rebel election commission, said Sunday that 350 polling stations were operating Sunday and that 1.4 million people were eligible to vote.

Critics argue the lax application of rules that oblige voters to be registered in the areas in which they cast their ballot could leave the way open to multiple voting or participation by non-locals.

A gunman dressed in military fatigues, who gave only his first name Alexander and the nom de guerre Raven, told AP at a polling station that despite being from Ukraine's Odessa region, he should be allowed to vote. Russian citizens fighting with the rebels should also be given that right, he said.

"They can fight and die here. So how come they can't vote here?" he said.

Five minutes later, Alexander successfully cast his ballot.

Turnout appeared to be brisk, in part due to the limited number of polling stations available. More than 200 people were seen lining up outside one polling station Sunday morning in the east of Donetsk.

Lyubov Khatsko, 55, who came to vote from the town of Marinka, just west of Donetsk, expressed despair at the continued unrest rocking her hometown.

"We have the right to have our own election, our freedom and to live the way we want to. We want the Ukrainians to get out of here," she said, speaking in Ukrainian.

The school where Khatsko cast her ballot was guarded by three armed rebels. An ammunition depot holding at least 20 automatic rifles was seen through an open door to a hall next to the room where the voting was taking place.

The election has been almost universally snubbed by international vote monitors, but a contingent of representatives from largely fringe Western and Russian political parties were observing the vote.

"As strange as it might sound, it is fine to hold elections in these conditions," said Alexei Zhuravlev, a lawmaker with Russia's Fatherland party. "This is the first step toward stabilizing the situation here. If we don't do it, this (war) will continue."