[Paper Review] Geometry Error Analysis of a parametric mapping for Higher Order Unfitted Space-Time Methods

Fabian Heimann, Christoph Lehrenfeld|arXiv (Cornell University)|Nov 4, 2023

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper presents a rigorous geometric error analysis of a parametric mapping for higher-order unfitted space-time finite element methods on moving domains. By introducing a computable mesh deformation (Θₕ) and comparing it to an ideal deformation (Ψ), the authors derive bounds on the approximation error in various norms and establish boundedness of higher-order derivatives, enabling optimal error estimates for unfitted space-time FEMs with geometric accuracy.

ABSTRACT

In [Heimann, Lehrenfeld, Preuß, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space-time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space-time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space-time finite element methods.

Motivation & Objective

• To develop a computationally feasible, higher-order accurate geometry description for time-dependent smooth domains in unfitted space-time finite element methods.
• To rigorously analyze the geometric approximation error between a computable mesh deformation (Θₕ) and an ideal deformation (Ψ) in space-time settings.
• To establish boundedness of the higher-order derivatives of the ideal parametric mapping Ψ, which is essential for optimal interpolation estimates.
• To validate the necessity of mesh refinement assumptions in the context of different blending strategies (FE vs. smooth blending).

Proposed method

• Introduce a space-time tensor-product background mesh with a levelset function ϕ to implicitly define moving domains.
• Define an ideal parametric mapping Ψ that exactly maps the reference element to the physical space-time domain.
• Construct a computable mesh deformation Θₕ using finite element interpolation of the levelset function and blending techniques.
• Apply different blending strategies—FE blending and smooth blending—to transition between deformed active elements and the exterior.
• Derive error bounds in L², H¹, and L∞ norms between Ψ and Θₕ, showing optimal convergence rates under appropriate assumptions.
• Analyze the regularity and boundedness of ∇Ψ and higher-order derivatives of Ψ, crucial for interpolation error estimates.

Experimental results

Research questions

• RQ1 How does the approximation error between the ideal mapping Ψ and the computable mapping Θₕ scale in different norms (L², H¹, L∞) under general space-time refinements?
• RQ2 What is the impact of different blending strategies (FE vs. smooth blending) on the boundedness and regularity of the computed deformation Θₕ?
• RQ3 Is the refinement restriction in Assumption 3.3 necessary to prevent unbounded spatial gradients in ∇Ψ?
• RQ4 To what extent does the geometry approximation error affect the convergence of space-time finite element methods?
• RQ5 How do the regularity and boundedness of Ψ’s higher-order derivatives influence the applicability of optimal interpolation estimates?

Key findings

• The approximation error between the ideal mapping Ψ and the computable mapping Θₕ is bounded in L², H¹, and L∞ norms, with optimal convergence rates under standard assumptions.
• The smooth blending strategy yields optimal convergence and boundedness of ∇Ψ, while the FE blending leads to a ∥∇Ψ∥∞ scaling as 1/h, indicating potential instability for fine meshes.
• The boundedness of ∇Ψ and higher-order derivatives of Ψ is established in Corollary 5.7, enabling the use of optimal space-time interpolation estimates.
• Numerical studies confirm that the FE blending causes ∥∇Ψ∥∞ to scale as 1/h under fine spatial refinements with fixed time steps, validating the necessity of Assumption 3.3.
• The interpolation error for Θₕ remains constant even as h → 0, but the gradient growth in Ψ suggests that the assumption on mesh refinement is essential for stability.
• The results in Theorem 5.8 provide rigorous proximity bounds between Ψ and Θₕ, forming a foundation for future error analysis of unfitted space-time FEMs.