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 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 $(u_1,{\mathbf{u}}_2)=(u,-{\nabla }_{\mathbf{x}}u)$. 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 ${\nabla }_{\mathbf{x}}{u}_1$ and ${\mathbf{u}}_2$, the (graph) norm of $U$ contains the $L_2$-norm of ${\partial }_t{u}_1+{\mathrm{div}}_{\mathbf{x}}{\mathbf{u}}_2$. 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 ${\mathbf{u}}_2$. 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 ${\partial }_t{u}_1+{\mathrm{div}}_{\mathbf{x}}{\mathbf{u}}_2$, i.e., of the forcing term $f=({\partial }_t-{\Delta }_x)u$. Numerical results show significantly improved convergence rates.

References in corpus (5)

• Further results on a space-time FOSLS formulation of parabolic PDEs
• A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations
• Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics
• Applications of a space-time FOSLS formulation for parabolic PDEs
• A well-posed First Order System Least Squares formulation of the instationary Stokes equations

Cited by in corpus (6)

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• An adaptive space-time method for nonlinear poroviscoelastic flows with discontinuous porosities
• Space-time FEM-BEM couplings for parabolic transmission problems
• Aubin--Nitsche-type estimates for space-time FOSLS for parabolic PDEs
• Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation