Opposition groups have alleged widespread voting irregularities
While the government hailed the poll in the Arab world's most populous country a success, opposition groups and rights bodies alleged widespread vote fraud.
The accusations included voter intimidation, ballot-stuffing, multiple voting, and election day campaigning.
Correspondents say turnout does not seem to have been high.
Officials said large crowds had been seen at some polling stations, but the BBC's Heba Saleh in Cairo said she did not see many people rushing out to vote.
Many Egyptians are disillusioned with politics after decades of authoritarian rule and state influence in the media, she says.
'Not fair'
Nine candidates are challenging Mr Mubarak.
One of the president's strongest opponents, Ayman Nour, said the elections had not been "not fair at all", and that he would reject rigged results.
I'm positive Hosni Mubarak will be re-elected, so why bother and vote?
Tina, Cairo, Egypt
Egypt elections: Your views Voter apathy marks poll
He said he believed only 15 to 20% of people voted in rural areas, and 3 to 5% in cities.
A senior official in the opposition Wafd party, Munir Abdel Nur, said: "I am very disappointed. I was not expecting a perfect election... but the extent of the irregularities and their premeditation was unacceptable."
Poll monitors cited irregularities, adding that voters in several areas had been bussed to polling stations and instructed to vote for Mr Mubarak.
There were also reports of other abuses, such as people being allowed to vote without voting cards and not being marked with indelible ink, or receiving money and goods in return for their votes.
ELECTION FACTS If no candidate polls 50%, the two with the highest number of votes proceed to round two Egyptians over 18 are required to vote by law. There are over 32m registered voters out of a population of some 74.9m Presidents elected for six year term
But an election commission spokesman defended the fairness of the vote, and predicted that turnout would be high.
This may have been a flawed election but, for many Egyptians, its main value is that it has established the principle of a competitive race for the top job, our correspondent says.
The US said the vote was "a beginning", and "a historic departure for Egypt".
Challengers
All 10 candidates were able to campaign freely and were given equal time on television.
But Mr Mubarak is the only leader the majority of Egyptians have known and most of his challengers are not even taken seriously as politicians.
Only two can be said to have constituencies.
One is Mr Nour of the Ghad (Tomorrow) party, an outspoken young liberal who says he would rule for two years only during which he would lay down the foundations of a democratic state.
His detention earlier this year earned Egypt public criticism from the US.
The other serious contender, Nomaan Gomaa, heads the Wafd, the once vigorous liberal party that fought for independence from Britain but that has been seriously weakened by the political restrictions of the past 50 years.
The biggest opposition group, the Muslim Brotherhood, is excluded from the election.
Terminadas las primeras elecciones presidenciales en Egipto con más de un candidato, son innumerables las denuncias de irregularidades. El principal problema radica en el control completo del partido del actual presidente Hosni Mubarak, según las organizaciones no gubernamentales que observaron el proceso. Esto permitió por ejemplo que se hiciera proselitismo en algunos centros de votación, que en algunos lugares se descuidara la instalación de un espacio privado para que los votantes eligieran su candidato, que los votantes del actual presidente recibieran regalos a la salida de los centros de votación, que se aceptaran votantes sin documentos de identificación y que se aceptara que en algunos centros hubiera más votos que votantes.

Será igualmente importante conocer el número de votantes dado que sólo recientemente los egipcios se han sentido motivados por participar en las votaciones. Estimaciones iniciales dan al principal candidato de oposición, Ayman Nour, hasta 15 % en las zonas rurales y hasta 5 % en las cuidades. Ayman Nour ha expresado que el número de irregularidades es tan alto que solicitará la invalidación del proceso.

Un representante de la Comisión Electoral Egipcia aseguró que los incidentes son mínimos y que la elección representará un hito en la democracia egipcia. Informó también que el conteo de los votos está en curso y que se emitirán resultados preliminares el viernes 9 de septiembre si hay una tendencia clara sobre el resultado.