Spline-based solution transfer for space-time methods in 2D+t

TL;DR

The paper addresses transferring solutions between nonconforming space-time slabs in slab-based finite element methods and the need for visualization-friendly and boundary-condition-ready interfaces. It introduces a spline-based transfer using HST-C (Hsieh-Clough-Tocher) splines to build a smooth surrogate at slab interfaces, followed by an $L_2$-projection to the next slab, with optional limiting to enforce a discrete maximum principle. The approach achieves $P_k$ exactness for $k \le 3$, demonstrates asymptotic mass conservation, and provides an error bound predicting second-order convergence for $1 \le k \le 3$, with numerical results showing third-order accuracy for $k=2$ on fine grids and improved visualization through the smooth surrogate. The method yields reliable visualization and boundary-condition enforcement in space-time simulations and offers potential extensions to higher dimensions and fully conservative adaptive quadrature strategies.

Abstract

This work introduces a new solution-transfer process for slab-based space-time finite element methods. The new transfer process is based on Hsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i) it maintains high-order accuracy up to 4th order, (ii) it preserves a discrete maximum principle, (iii) it asymptotically enforces mass conservation, and (iv) it constructs a smooth, continuous surrogate solution in between space-time slabs. While many existing transfer methods meet the first three requirements, the fourth requirement is crucial for enabling visualization and boundary condition enforcement for space-time applications. In this paper, we derive an error bound for our HCT spline-based transfer process. Additionally, we conduct numerical experiments quantifying the conservative nature and order of accuracy of the transfer process. Lastly, we present a qualitative evaluation of the visualization properties of the smooth surrogate solution.

Figures (17)

• Figure 1: Transferring the solution $v_{h_a}$ on the terminating surface mesh $\mathcal{T}_a$ of the space-time slab $Q_n$ to the initial surface mesh $\mathcal{T}_b$ of the subsequent space-time slab $Q_{n+1}$. The smoothed solution is obtained using the smoothing operator $E(\cdot)$ on the solution $v_{h_a}$. This solution $E(v_{h_a})$ provides a surrogate solution at $t_n$ that is suitable for visualization and boundary condition enforcement. Projection operator $I_{h_b}(\cdot)$ transfers the smoothed solution to the initial grid of $Q_{n+1}$.
• Figure 2: Left, the degrees of freedom for a triangular element with $k = 1$. Middle, the degrees of freedom for a triangular element with $k = 2$. Right, the degrees of freedom for a triangular element with $k = 3$.
• Figure 3: The synchronization procedure for a discontinuous finite element solution, for the $k=1$ case.
• Figure 4: The HCT-C triangle as defined by Bernadou and Hassan bernadou1981basis, shown with its degrees of freedom, $\Sigma_T$.
• Figure 5: Left, triangle $T_m$ and three subtriangles formed by $o$, where $o$ is inside $T_m$. Right, triangle $T_m$ and three subtriangles formed by $o$, where $o$ is outside of $T_m$.

Theorems & Definitions (9)

• Definition 3.2: $H^2$ Smoothing Operator
• Lemma 3.3: Bound on $H^2$ Smoothing Operator
• proof
• Definition 3.4: Projection Operator
• Lemma 3.5: Bound on Projection Operator
• proof
• Theorem 3.6: Error Estimate
• proof
• Remark 3.1