How do adult differentiated somatic cells divide?
There is no known alternative mechanism of proper cell division apart from mitosis and meiosis in animals. I think this is an unfortunate phrasing where the author is trying to introduce the term "postmitotic". The reason for saying "most  adult differentiated cells" is because some cells in adults are considered differentiated but they still proliferate under certain conditions, like for example the cells of the adaptive immune system. 

It could also be that the author is trying to be overly cautious because there are cells that split parts of themselves off and release them. These mechanisms could be considered cell division in some sense but are neither mitosis nor meiosis. Examples of this are the release of platelets by megakaryocytes and the release of extracellular vesicles by cells undergoing apoptosis.