Title: On a space-time first-order system least-squares formulation of parabolic PDEs
Abstract:
While the common space-time variational formulation of a parabolic equation results in a bilinear form that is non-coercive, [1] recently proved well-posedness of a space-time first-order system least-squares (FOSLS) formulation of the heat equation, which corresponds to a symmetric and coercive bilinear form. In particular, the Galerkin approximation from any conforming trial space exists and is a quasi-optimal approximation. Additionally, the least-squares functional automatically provides a reliable and efficient error estimator.
In [2], we have generalized this least-squares method to general second-order parabolic PDEs with possibly inhomogenoeus Dirichlet or Neumann boundary conditions. For homogeneous Dirichlet conditions, we present convergence of adaptive finite element methods driven by the built-in least-squares estimator. As the convergence rates can still be slow for highly singular solutions, we have constructed a trial space with enhanced approximation properties. Moreover, we employ the space-time least-squares method for optimal control problems and finally present a generalization to the instationary Stokes equations with slip boundary conditions.
References:
[1] Führer, Karkulik, Space–time least-squares finite elements for parabolic equations, CAMWA (2021)
[2] Gantner, Stevenson, Further results on a space-time FOSLS formulation of parabolic PDEs, M2AN (2021)
