Improved rates for a space–time FOSLS of parabolic PDEs

Abstract

AbstractWe 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 $$(u_1,\textbf{u}_2)=(u,-\nabla _\textbf{x} u)$$ ( u 1 , u 2 ) = ( u , - ∇ x u ) . The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of $$L_2$$ L 2 -type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides $$L_2$$ L 2 -norms of $$\nabla _\textbf{x} u_1$$ ∇ x u 1 and $$\textbf{u}_2$$ u 2 , the (graph) norm of U contains the $$L_2$$ L 2 -norm of $$\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$$ ∂ 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 $$\textbf{u}_2$$ 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 $$\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2$$ ∂ t u 1 + div x u 2 , i.e., of the forcing term $$f=(\partial _t-\Delta _x)u$$ f = ( ∂ t - Δ x ) u . Numerical results show significantly improved convergence rates.