Improved rates for a space–time FOSLS of parabolic PDEs

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components (u₁, u₂) = (u, - ∇_xu). The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L₂-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L₂-norms of ∇_xu₁ and u₂, the (graph) norm of U contains the L₂-norm of ∂_tu₁ + div_xu₂. When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of u₂. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of ∂_t, i.e., of the forcing term ƒ = (∂_t - Δ_x). Numerical results show significantly improved convergence rates.