Improved rates for a space-time FOSLS of parabolic PDEs
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We consider the first-order system space-time formulation of the heat equation introduced in [Bochev, Gunzburger, Springer, New York (2009)], and analyzed in [Führer, Karkulik, Comput. Math. Appl. 92 (2021)] and [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal. 55 (2021)], with solution components (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-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L_2-norms of ∇_ x u_1 and u_2, the (graph) norm of U contains the L_2-norm of ∂_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 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 u_1 + div_ x u_2, i.e., of the forcing term f=(∂_t-Δ_x)u. Numerical results show significantly improved convergence rates.
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