Since initial work by Satoshi, who suggested proof of work consensus schema, that implies mining, many other distributed consensus schemas were invented. Some of these schemas are applicable to public trustless blockchain, while the other may only be used in private trustful ones. Here trustful means that all participants trust small group of validators who responsible for establishing transaction ordering and decide which one of several conflicting transactions will be finally included into blockchain.
For example, J.P.Morgan Quorum private blockchain uses Istanbul BFT consensus that requires modifications in
geth. Out of the box,
geth supports proof of authority consensus, currently used by Rinkeby test network.
In general, mining is needed to make all nodes agree on who is eligible to construct next block. So, for permissioned blockchains, there are usually small number of validators knowing each other in real life, who may negotiate and agree on the contents of next block without necessity to spend lots of resource on mining. The other nodes will trust blocks as long as they have enough validator signatures.