A group of frustrated supporters gathered outside the ground to call for manager Paul Groves to be sacked and demanded that Mitchell show his face.
Mitchell then appeared but was shouted down as he tried to address the crowd, with one Cherries fan insisting he would give his season ticket back if Groves was not sacked. The south-coast club are 18th in League One, having won just once all season.
"It's all very well changing it, but we have to find a way to win. I could have employed Glenn Hoddle but we might still have been in the same situation," Mitchell told supporters. "It's football. The ball is round and we are trying our best."
Shouting and swearing came from both Mitchell and some of the Bournemouth fans as the anger and chants for Groves to go became more heated, before the Cherries chairman responded to one individual by saying: "You haven't got any information have you? You haven't got a clue."
Mitchell was then hit in the back of the head by a matchday programme, which had been thrown by one of the people gathered, as he turned to go back into the reception of the ground.