Improved rates for a space-time FOSLS of parabolic PDEs

TL;DR

The paper advances a space-time first-order system least-squares (FOSLS) discretization for parabolic PDEs by using prismatic space-time elements to overcome reduced convergence for non-smooth data. It constructs a local quasi-interpolant with a near-commuting diagram, and then forms a global conforming space via averaging, enabling interpolation error bounds that depend primarily on the forcing term’s regularity. Numerical experiments in 1+1D and 2+1D with nonmatching initial data and interior/boundary singularities show substantially improved convergence rates for prismatic meshes over traditional simplicial meshes, especially under adaptive refinement. The results suggest that prismatic space-time discretizations, possibly with anisotropic temporal-spatial meshing, offer robust, efficient solutions for non-smooth parabolic problems in practice.

Abstract

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 $(u_1,{\bf u}_2)=(u,-\nabla_{\bf x} u)$. 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 $\nabla_{\bf x} u_1$ and ${\bf u}_2$, the (graph) norm of $U$ contains the $L_2$-norm of $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$. 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 ${\bf u}_2$. 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 $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$, i.e., of the forcing term $f=(\partial_t-Δ_x)u$. Numerical results show significantly improved convergence rates.

Figures (6)

• Figure 1: Convergence plot of Section \ref{['sec:2d_f12_u01']} for spatial domain $\Omega=(0,1)$, right-hand side $(f_1,{\bf f}_2,u_0) = (2,0,1)$, simplicial and prismatic meshes, and uniform and adaptive refinement.
• Figure 2: Convergence plot of Section \ref{['sec:2d_f10_u0hat']} for spatial domain $\Omega=(0,1)$, right-hand side $(f_1,{\bf f}_2,u_0) = (1,0,x\mapsto 1-2|x-\tfrac{1}{2}|)$, simplicial and prismatic meshes, and uniform and adaptive refinement.
• Figure 3: Convergence plot of Section \ref{['sec:2d_f10_u0x05']} for spatial domain $\Omega=(0,1)$, right-hand side $(f_1,{\bf f}_2,u_0) = (0,0,x\mapsto x^{1/2}(1-x))$, simplicial and prismatic meshes, and uniform and adaptive refinement.
• Figure 4: Convergence plot of Section \ref{['sec:3d_f10_u01']} for spatial domain $\Omega=(0,1)^2$, right-hand side $(f_1,{\bf f}_2,u_0) = (0,0,1)$, simplicial and prismatic meshes, and uniform and adaptive refinement.
• Figure 5: Convergence plot of Section \ref{['sec:3d_f0_u0r1']} for spatial domain $\Omega=(0,1)^2$, right-hand side $(f_1,{\bf f}_2,u_0) = (0,0, {\bf x}\mapsto \sqrt{(x_1-\tfrac{1}{2})^2+(x_2-\tfrac{1}{2})^2} \sin(\pi x_1) \sin(\pi x_2))$, simplicial and prismatic meshes, and uniform and adaptive refinement.

Theorems & Definitions (14)

• Remark 1.1
• Remark 2.1
• Proposition 2.2: Commuting diagram property
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Corollary 2.5
• Remark 2.6
• Proposition 3.1