7. Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution

In this work, we present an overview on the development of space–time finite element methods for the numerical solution of some parabolic evolution equations with the heat equation as a model problem. Instead of using more standard semi– discretization approaches such as the method of lines or Rothe’s method, our specific focus is on continuous space–time finite element discretizations in space and time simultaneously. While such discretizations bring more flexibility to the space–time finite element error analysis and error control, they usually lead to higher computational complexity and memory consumptions in comparison with standard time– stepping methods. Therefore, progress on a posteriori error estimation and respective adaptive schemes in the space–time domain is reviewed, which aims to save a number of degrees of freedom, and hence reduces complexity, and recovers optimal order error estimates. Further, we provide a summary on recent advances in efficient parallel space–time iterative solution strategies for the related large–scale linear systems of algebraic equations, that are crucial to make such all–at–once approaches competitive with traditional time stepping methods. Finally, some numerical results are given to demonstrate the benefits of a particular adaptive space–time finite element method, the robustness of some space–time algebraic multigrid methods, and the applicability of space–time finite element methods for the solution of some parabolic optimal control problem.