10 December – Gregor Gantner: Applications of a space-time first-order system least-squares formulation for parabolic PDEs.

Gregor Gantner Friday 10th December at 11:00



Currently, there is a growing interest in simultaneous space-time methods for solving parabolic evolution equations. The main reasons are that, compared to classical time-marching methods, space-time methods are much better suited for massively parallel implementation, are guaranteed to give quasi-optimal approximations from the employed trial space, have the potential to drive optimally converging simultaneously space-time adaptive refinement routines, and they provide enhanced possibilities for reduced order modelling of parameter-dependent problems. On the other hand, space-time methods require more storage. This disadvantage however vanishes for problems of optimal control, for which the solution is needed simultaneously over the whole time interval anyway.

While the common space-time variational formulation of a parabolic equation results in a bilinear form that is non-coercive, [1] recently proved the well-posedness of a space-time first-order system least-squares formulation of the heat equation. Least-squares formulations always correspond to a symmetric and coercive bilinear form. In particular, the Galerkin approximation from any conforming trial space exists and is a quasi-best approximation. Additionally, the least-squares functional automatically provides a reliable and efficient error estimator.

In [2], we have generalized the least-squares method of [1] to general second-order parabolic PDEs with possibly inhomogeneous Dirichlet or Neumann boundary conditions.
For homogeneous Dirichlet conditions, we present in this talk convergence of a standard adaptive finite element method driven by the least-squares estimator, which has also been demonstrated in [2].
The convergence analysis is applicable to a wide range of least-squares formulations for other PDEs, answering a long-standing open question in the literature.
Moreover, we employ the space-time least-squares method for parameter-dependent problems as well as optimal control problems.
In both cases, the coercivity of the corresponding bilinear form plays a crucial role.

[1] T. Führer and M. Karkulik. Space–time least-squares finite elements for parabolic equations. Comput. Math. Appl., 92:27–36, 2021.

[2] G. Gantner and R. Stevenson. Further results on a space-time FOSLS formulation of parabolic PDEs. ESAIM Math. Model. Numer. Anal., 55(1):283–299, 2021.

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