Would you like to solve a partial differential equation efficiently with a relative error of 10% or would you prefer to wait a bit longer and solve it with an error of only 1% ? Is it sufficient to know that the error is about 1% (having no idea what the `about' means) or would you prefer to have reliable information that the error is guaranteed to be below the required tolerance?
Answering these questions is necessary for the efficient and reliable numerical solution of practically any mathematical problem. In the context of numerical solution of partial differential equations, the crucial tool is the adaptive algorithm with suitable error indicators and estimators. I will overview the adaptive algorithm and its variants. I will concentrate on the a posteriori error estimators with the emphasis on the guaranteed ones.