Unfitted Space-Time FEM for Evolving Domains

Updated 3 September 2025

Unfitted space-time finite element methods are numerical techniques that discretize time-dependent PDEs on evolving domains using fixed background meshes and implicit level-set representations.
They employ stabilization strategies such as ghost penalty and aggregation constraints to manage small cut cells and ensure robust, high-order convergence.
Applications include solving advection-diffusion on moving manifolds, coupled bulk-surface problems, and multiphysics simulations involving large deformations and topological changes.

The unfitted space-time finite element method (USTFEM) refers to a broad class of numerical techniques for the discretization of time-dependent partial differential equations (PDEs) posed on evolving, possibly complex, geometries using finite element spaces that are defined independently of (and not aligned with) the domain boundary or surface at each time instance. These methods combine classical space-time variational principles with unfitted, cut, or embedded finite element concepts to handle moving domains, interfaces, or surfaces with arbitrary complexity, large deformation, and topological change, all without remeshing.

1. Core Principles and Terminology

USTFEM leverages a fixed background mesh (bulk mesh or reference mesh) on a computational domain that strictly contains all possible spatial configurations of the evolving physical domain $\Omega(t)$ . The moving domain or interface is described implicitly—typically by a level set function $\phi(x,t)$ —and the active (physical) domain at each time $t$ is given as $\Omega(t)=\{x : \phi(x,t) < 0\}$ . The space-time regions traced by $\Omega(t)$ (and possibly a lower dimensional moving manifolds $\Gamma(t)$ ) then define the geometry on which the PDE must be solved.

Key abstractions in USTFEM include:

Bulk finite element spaces: Finite element spaces $V_h$ of prescribed smoothness defined on the background mesh $\mathcal{T}_h$ .
Cut elements: Mesh elements that are intersected by the domain boundary.
Isoparametric mappings: Nonlinear, element-wise polynomial mappings $\Theta_h^{st}$ to reconstruct higher-order accurate geometry from a piecewise linear approximation of $\phi$ (Heimann et al., 11 Apr 2025, Heimann et al., 2022).
Trace or embedded FE spaces: Restrictions or traces of bulk functions on the (possibly evolving) physical domain, with appropriate constraints to ensure approximation and continuity.

Stabilization terms—including ghost penalty (jump penalization on faces near cut regions), extension operators, or aggregation constraints—are universally included to guarantee the well-posedness and robustness of the unfitted discretization in the presence of small cut cells (Lehrenfeld et al., 2018, Heimann et al., 2022, Badia et al., 2022).

2. Discretization Strategies

Space-time discretizations can be implemented in several interrelated ways:

Time-slab and tensor-product variational approaches: The global space-time domain is partitioned into slabs $I_n = (t_{n-1}, t_n]$ and tensor-product finite element spaces $V_h^{(k_s)} \otimes \mathcal{P}^{(k_t)}(I_n)$ are employed on the "active" cut mesh for each slab (Burman et al., 2 Sep 2025, Heimann et al., 2022, Lou et al., 2021).
Discontinuous and continuous Galerkin in time: Discontinuous Galerkin (dG) methods admit jump terms on time interfaces; continuous Galerkin (cG) and Galerkin-collocation variants enforce varying levels of temporal continuity and sometimes enforce continuity of time derivatives up to a specified order (Heimann et al., 2022).
Eulerian and interface capturing: The domain and its evolution are handled completely in the Eulerian reference frame, with all quantities defined on the background mesh and transported across time intervals using extension/projection operators and stabilization (Lehrenfeld et al., 2018, Lou et al., 2021).
Higher-order, isoparametric geometry handling: Geometry is approximated with high-order accuracy in both space and time using mapping $\Theta_h^{st}$ , ensuring that the geometric errors affecting quadrature and basis function representation are of higher order and do not pollute the overall convergence (Heimann et al., 11 Apr 2025, Reusken et al., 2 Jan 2024, Heimann, 15 Jan 2024).

In a typical formulation, the evolution law is discretized in a weak form on a domain with boundaries approximated at each time slab and integrals computed using high-order quadrature adapted to the cut geometry.

3. Stabilization, Extension, and Robustness

The unfitted nature of USTFEM leads to cut elements whose intersection with the physical domain can be arbitrarily small, causing severe conditioning issues unless addressed.

Ghost penalty stabilization: Additional penalty terms are introduced over interior facets near the physical boundary, controlling the jump of traces of basis functions (and potentially their derivatives) across elements (Lehrenfeld et al., 2018, Heimann et al., 2022, Burman et al., 2 Sep 2025). These terms act as discrete extension operators, ensuring that degrees of freedom (DOFs) that become "active" as the boundary moves are properly controlled.
Aggregate FE/constraint-based stabilization: Problematic (badly cut) DOFs are explicitly constrained as linear combinations of well-behaved ("root") DOFs to form the aggregated FE space, preserving optimal approximation properties and ensuring well-condition of the linear system (Verdugo et al., 2019, Badia et al., 2020, Badia et al., 2022).
Projection/extension operators in time: In the presence of moving domains that change the set of active DOFs from one time step to the next, operators are introduced to transfer (project or extend) the solution from the previous time slab to the current one in a stable and accurate fashion (Lehrenfeld et al., 2018, Lou et al., 2021, Heimann et al., 11 Apr 2025).

Mechanisms are provided to guarantee mass conservation (especially for conservative or advection-diffusion problems): for instance, hybrid schemes using projection and ghost penalty stabilization, or by enforcing conservation constraints via Lagrange multipliers (Deckelnick et al., 2013).

4. Geometric Approximation and Error Analysis

The accuracy and computational efficiency of USTFEM are strongly linked to the treatment of the geometry:

Level set approximation: The implicit surface or volume boundary is represented by a level-set function, and numerical schemes assume polynomial degree $q_s$ in space and $q_t$ in time for the discrete level-set $\phi_h$ .
Isoparametric mapping errors: Errors between the true domain $Q$ and the discrete isoparametric domain $Q^h$ are quantified by estimates

$\|\partial_t^{l_t} D_x^{l_s}(Id - \Phi^{st})\|_{L^\infty(\widetilde{Q} \times I_n)} \lesssim h^{q_s+1-l_s} + \Delta t^{q_t+1-l_t}$

for $l_s + l_t \in \{0, 1\}$ , ensuring that the geometric consistency errors appear as higher-order terms in the overall error bound (Heimann et al., 11 Apr 2025).

Stability and error bounds: The error analysis typically employs Strang-type lemmata, splitting the total error into approximation, geometric consistency, and stability terms. An inf-sup or energy stability analysis is derived for the complete system, including the geometric errors, leading to optimal $h^k$ and $\Delta t^r$ rates under appropriate regularity of the solution and the geometry (Heimann et al., 2022, Heimann et al., 11 Apr 2025).
A priori error estimates: For multidimensional and time-dependent domains, error bounds in problem-specific norms (such as DG-norms with material derivative contributions) are established, confirming high-order convergence for both the solution and geometric observables (Reusken et al., 2 Jan 2024, Burman et al., 2 Sep 2025).

5. Applications and Numerical Performance

USTFEM has been demonstrated for a broad array of problems involving complex temporal and spatial domain evolution:

Surface PDEs: Elliptic and parabolic PDEs on evolving hypersurfaces, using surface integral weak formulations, as in advection-diffusion on moving manifolds. Mass-preserving formulations for advection-diffusion laws on evolving surfaces are a haLLMark (Deckelnick et al., 2013).
Bulk and coupled surface-bulk problems: Scalar transport, convection-diffusion, and coupled bulk-surface systems with interface conditions—including biological cell models and multiphase flow—are routinely treated (Heimann, 15 Jan 2024).
Variable topology and large deformation: The use of an underlying fixed mesh and a level-set representation allows simulation of domains undergoing arbitrary deformations or topological transitions (merging/splitting), observed in mass transfer, viscous flow, and fluid-structure interaction (Lehrenfeld et al., 2018, Badia et al., 2022, Heimann et al., 2022).
Wave propagation and hyperbolic problems: Explicit time integration, strong stability, and hp-convergence are achieved for wave equations and transport with discontinuous coefficients or evolving interfaces (Chen et al., 2021, Burman et al., 2 Sep 2025).

Numerical experiments robustly show that with appropriate stabilization and isoparametric geometry handling, USTFEM achieves optimal convergence rates in $L^2$ , $H^1$ , and problem-specific norms, even in the presence of complex geometry evolution. Large-scale parallelization and adaptive refinement (via tree-based and forest-of-trees approaches) are practical, extending applicability to multi-million DOF problems (Verdugo et al., 2019, Badia et al., 2020).

USTFEM can be contrasted with:

Classical ALE (Arbitrary Lagrangian-Eulerian) schemes: ALE methods require mesh movement or remeshing to conform to the moving domain, while USTFEM operates solely on a fixed background mesh, providing easier handling of topological events and large deformations.
XFEM/embedded/immersed approaches: These also use unfitted discretizations, but USTFEM integrates the time dimension variationally and treats geometry consistently at high order in both space and time.
Fictitious domain, finite cell, and CutFEM methodologies: USTFEM extends and generalizes these methods to the space-time, moving geometry context, with particular emphasis on stabilization/aggregation and higher-order geometry treatment (Heimann et al., 11 Apr 2025, Heimann et al., 2022).

Key innovations of USTFEM documented in the literature include robust handling of small cut cells via aggregation and ghost penalties, isoparametric handling of high-order geometry errors, rigorous energy and inf-sup stability in problem-specific norms, and efficient parallel/distributed implementations (Verdugo et al., 2019, Badia et al., 2022, Reusken et al., 2 Jan 2024).

7. Outlook and Open Problems

Research in USTFEM is progressing toward:

Highly efficient solvers: Scalable distributed-memory solvers (multigrid, Krylov methods) exploiting the robust conditioning conferred by aggregation and stabilization (Verdugo et al., 2019).
Adaptive refinement: Integration of functional and geometric error indicators for dynamic space-time adaptivity (Badia et al., 2020).
Multiphysics and multiphase flows: Extension of USTFEM to coupled systems (e.g., Navier-Stokes on moving domains, bulk-surface reaction-diffusion) and problems with evolving or reactive interfaces.
Analysis for low-regularity and nonsmooth interfaces: Error estimates for cases with Lipschitz or even singular geometry evolution.

Unfitted space-time finite element methods present a mathematically and computationally comprehensive framework for the simulation of PDEs on evolving domains and nonstationary surfaces, with the key advantages of higher-order accuracy, flexibility in treating complex and changing geometry, and robust numerical stability and scalability (Burman et al., 2 Sep 2025, Heimann et al., 11 Apr 2025, Heimann et al., 2022).