Improved rates for a space-time FOSLS of parabolic PDEs
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arXiv
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| Formato: | Preprint |
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2022
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| _version_ | 1866910347666915328 |
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| author | Gantner, Gregor Stevenson, Rob |
| author_facet | Gantner, Gregor Stevenson, Rob |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10824 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improved rates for a space-time FOSLS of parabolic PDEs Gantner, Gregor Stevenson, Rob Numerical Analysis 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. |
| title | Improved rates for a space-time FOSLS of parabolic PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2208.10824 |