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

Olaf Steinbach and Huidong Yang

Abstract

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 semidiscretization 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. Whereas 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 timestepping 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.

© 2019 Walter de Gruyter GmbH, Berlin/Munich/Boston
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