Improved rates for a space–time FOSLS of parabolic PDEs

Abstract 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, - ₓ u) (u 1, u 2) = (u, - ∇ x u). The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L₂ L 2 -type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L₂ L 2 -norms of ₓ u₁ ∇ x u 1 and u₂ u 2, the (graph) norm of U contains the L₂ L 2 -norm of ₜ u₁ +{\, div\, }ₓ u₂ ∂ t u 1 + div x u 2. 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 u 2. 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 u₁ +{\, div\, }ₓ u₂ ∂ t u 1 + div x u 2, i. e. , of the forcing term f= (ₜ- ₓ) u f = (∂ t - Δ x) u. Numerical results show significantly improved convergence rates.