Analysis of the continuous-in-time space-time unfitted FEM

The isoparametric unfitted space-time finite element framework developed in the group supports multiple time-discretization strategies: discontinuous Galerkin (DG) in time, the continuous Petrov–Galerkin (cPG) method, and Galerkin collocation. While numerical experiments validate all three approaches, a rigorous a priori error analysis currently exists only for the DG-in-time case. This project derives stability and optimal-order error estimates for the continuous-in-time (cPG) variant.

Background: the space-time framework

Consider a convection-diffusion problem on a moving domain $\Omega(t) \subset \mathbb{R}^d$, $t \in [0,T]$:

$\partial_t u + \mathbf{w}\cdot\nabla u - \mu\Delta u = f \quad\text{in }\Omega(t),\quad u = 0\text{ on }\partial\Omega(t),$

where $\mathbf{w}$ is the domain velocity. In the space-time formulation, the space-time domain $\mathcal{Q} = \bigcup_{t\in[0,T]}\Omega(t)\times\{t\}$ is embedded in a background space-time mesh $\mathcal{T}_h^{st}$ (a tensor product of a spatial mesh with a time partition). The isoparametric mapping $\Phi_h$ deforms mesh elements to accurately approximate $\partial\mathcal{Q}$.

In [HLP2023] (Heimann, Lehrenfeld, Preuß, SISC 2023), the continuous Petrov–Galerkin (cPG) method is introduced alongside the DG-in-time scheme. For cPG, the trial space $V_h^k$ consists of functions that are continuous in time and polynomial of degree $k$ on each space-time slab, while the test space $W_h^{k-1}$ consists of functions piecewise polynomial of degree $k-1$ with no inter-slab continuity. The bilinear form is:

$B(u_h, v_h) = \int_{\mathcal{Q}_h} (\partial_t u_h + \mathbf{w}\cdot\nabla u_h)\,v_h + \mu\nabla u_h\cdot\nabla v_h + j_h(u_h, v_h),$

where $j_h$ is the ghost-penalty stabilization. Compared to the DG-in-time method, cPG avoids the inter-slab jump terms and gives a smaller global system for the same polynomial degree.

What is known and what is open

The analysis of the DG-in-time space-time unfitted method has been completed in two steps:

• [HL_IMAJNA_2025] (Heimann, Lehrenfeld, IMA J. Numer. Anal. 2025): geometry error bounds — how well does $\Phi_h$ approximate the true space-time interface?
• [HLP_IMAJNA_2025] (Heimann, Lehrenfeld, Preuß, IMA J. Numer. Anal. 2025): full stability and error estimates for the DG-in-time case, accounting for geometry errors from [HL_IMAJNA_2025].

The analysis for the cPG case is open. The main challenges are:

• Inf-sup stability: the standard coercivity argument for the DG method (using the numerical flux jump) is replaced by a Petrov–Galerkin inf-sup condition, which requires different techniques (e.g. a local inverse inequality relating the cPG trial space to the test space).
• Interaction with ghost penalty: the ghost-penalty term in [HLP_IMAJNA_2025] is designed to stabilize the DG inter-slab jumps; for cPG it must be re-examined since there are no jumps.
• Geometry perturbation: the geometry error analysis [HL_IMAJNA_2025] carries over essentially unchanged, but the way it enters the error estimate must be rederived for the cPG bilinear form.

Project tasks

1. Study the existing DG-in-time analysis. Work through [HLP_IMAJNA_2025] in detail to understand the proof structure: stability via ghost-penalty norms, approximation via Scott–Zhang interpolation, and how geometry errors enter via [HL_IMAJNA_2025].
2. Inf-sup stability for the cPG method on a fixed domain. As a warm-up, prove inf-sup stability of the cPG method (without the unfitted setting) on a fixed domain. This uses an inverse inequality relating $\|\partial_t u_h\|$ in the test norm to $\|u_h\|$ in the trial norm.
3. Ghost-penalty norm and the cPG extension property. Derive appropriate ghost-penalty norms for the cPG trial space. Show that the extension property (degrees of freedom are well-defined even for unfavorable cuts) holds under ghost-penalty stabilization.
4. Inf-sup stability in the unfitted setting. Extend the fixed-domain stability to the unfitted space-time case, accounting for the isoparametric mapping and the geometry perturbation.
5. A priori error estimate. Combine inf-sup stability with interpolation estimates (Scott–Zhang in space-time) and the geometry error bounds from [HL_IMAJNA_2025] to derive optimal-order error estimates in the energy norm.
6. Numerical verification. Implement the cPG method in NGSolve / ngsxfem and verify the predicted convergence rates (already done numerically in [HLP2023]; here the focus is on confirming the theoretical constants and the sharpness of the estimates).

References

• HLP2023 — Heimann, Lehrenfeld, Preuß: Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains (SISC 2023) — introduces cPG variant numerically
• HL_IMAJNA_2025 — Heimann, Lehrenfeld: Geometry Error Analysis of a parametric mapping for Higher Order Unfitted Space-Time Methods (IMA J. Numer. Anal. 2025)
• HLP_IMAJNA_2025 — Heimann, Lehrenfeld, Preuß: Discretization Error Analysis of a High Order Unfitted Space-Time Method for moving domain problems (IMA J. Numer. Anal. 2025) — DG-in-time analysis
• LO_ESAIM_2019 — Lehrenfeld, Olshanskii: Eulerian FEM for PDEs in time-dependent domains (ESAIM 2019) — foundational BDF approach
• LL_SINUM_2022 — Lou, Lehrenfeld: Isoparametric unfitted BDF–FEM for PDEs on evolving domains (SINUM 2022) — higher-order BDF analysis