Geometry Error Analysis of a parametric mapping for Higher Order Unfitted Space-Time Methods

TL;DR

The paper addresses geometry errors in higher-order unfitted space-time FEM for moving domains by developing a parametric space-time mapping framework. It introduces an ideal space-time mapping on active elements and constructs computable, continuous discrete mappings via semi-discrete and fully discrete steps, using both FE and smooth blending. Key contributions include rigorous bounds for the distance between the realized and ideal mappings in space and time, regularity estimates, and extension procedures that yield near-identity deformations on the full space-time mesh. Numerical experiments confirm geometrical accuracy and interpolation quality, and the results pave the way for rigorous error analyses of unfitted space-time discretisations for bulk and surface problems.

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.

Figures (12)

• Figure 1: Sketch of a simple subdivision for quadrature on $T \cap \Omega^{\text{lin}}$, the discrete geometry $\Omega^{\text{lin}}$ and the mapped geometry $\Omega^h$ obtained from the mapping $\Theta_h$. Reproduction from HLP2022.
• Figure 1: Illustration of the space-time geometry obtained from a circle moving in a square background domain. The left figure shows the space-time domain $Q$ embedded in a set of three time slabs, and the right figure shows the tensor-product background space-time mesh of three time slabs.
• Figure 1: Illustration of the discrete regions involved in the FE blending construction. We sketch the situation of a large (left) and small (right) time step. The solid and transparent red lines indicate the discrete interface at the beginning and end of the time step, respectively. The blue elements indicate the active mesh as it relates to the blending construction, \ref{['def:Thb1']}. The green elements indicate the additional adjacent elements where the FE blending will operate.
• Figure 1: Outline of the construction and analysis of the mappings on active elements. The ideal mapping is on the top left corner of the diagram while the realizable mapping used in practice is on the bottom right corner. The intermediate mappings allow for a step-by-step analysis of the error (in different norms) between the ideal and the realizable mapping.
• Figure 1: Upper row left and middle: Two examples of blending functions \ref{['def:b']} at a fixed time for increasing values of $w_b$. Upper row right: Examples of the function $\pi_s$ involved in the definition of \ref{['def:b']} in terms of $\phi$, \ref{['eq:def_b_example']}. Bottom row left and middle: Absolute value of the deformation stemming from the smooth blending functions respectively above. Bottom row right: Absolute value of the deformation from the \ref{['def:FE']} blending for the same discretisation parameters and a small time step.

Theorems & Definitions (72)

• Definition 2.2
• Remark 2.7
• Corollary 2.8
• Corollary 2.9
• Remark 3.1: Problems purely posed on space-time cut elements
• Remark 3.6: Comparison of blending options
• Lemma 4.1
• Proof 1
• Corollary 4.2
• Lemma 4.3