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 \((u_1,\textbf{u}_2)=(u,-\nabla _\textbf{x} u)\). The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of \(L_2\)-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides \(L_2\)-norms of \(\nabla _\textbf{x} u_1\) and \(\textbf{u}_2\), the (graph) norm of U contains the \(L_2\)-norm of \(\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2\). 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 \(\textbf{u}_2\). 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 \(\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2\), i.e., of the forcing term \(f=(\partial _t-\Delta _x)u\). Numerical results show significantly improved convergence rates.