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Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements

Erik Burman, Janosch Preuss

TL;DR

This work develops a stabilized unfitted isoparametric finite element method for unique continuation problems across interfaces with jump coefficients. By combining an isoparametric geometry resolution with Nitsche-type interface terms, CIP/Gaussian least-squares stabilization, and a weak Tikhonov-like bulk term, the authors derive error estimates that account for geometric errors via the mapping $\Phi_h$ and conditional stability in the data-to-target extension. The main contributions are a complete error analysis showing convergence in the stabilized norm and $L^2(B)$-error bounds driven by the Hölder exponent $\tau$ from conditional stability, plus extensive numerical experiments illustrating the influence of geometry accuracy, wavenumber, and coefficient contrasts on stability and accuracy. The method demonstrates competitive performance with fitted schemes, while enabling high-order geometry handling and applicability to inverse-type continuation problems in subsurface and seismic contexts. The results inform practical use of unfitted methods for ill-posed interface problems and open avenues for robust preconditioning and advanced geometry techniques.

Abstract

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber.

Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements

TL;DR

This work develops a stabilized unfitted isoparametric finite element method for unique continuation problems across interfaces with jump coefficients. By combining an isoparametric geometry resolution with Nitsche-type interface terms, CIP/Gaussian least-squares stabilization, and a weak Tikhonov-like bulk term, the authors derive error estimates that account for geometric errors via the mapping and conditional stability in the data-to-target extension. The main contributions are a complete error analysis showing convergence in the stabilized norm and -error bounds driven by the Hölder exponent from conditional stability, plus extensive numerical experiments illustrating the influence of geometry accuracy, wavenumber, and coefficient contrasts on stability and accuracy. The method demonstrates competitive performance with fitted schemes, while enabling high-order geometry handling and applicability to inverse-type continuation problems in subsurface and seismic contexts. The results inform practical use of unfitted methods for ill-posed interface problems and open avenues for robust preconditioning and advanced geometry techniques.

Abstract

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber.
Paper Structure (30 sections, 1 theorem, 131 equations, 9 figures)

This paper contains 30 sections, 1 theorem, 131 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. The problem setting
  4. Content and structure of the article
  5. Resolving the interface by an isoparametric mapping
  6. The exact geometry
  7. The piecewise linear reference geometry
  8. The isoparametric mapping
  9. Basic properties of the mappings
  10. Geometry errors for mapped functions
  11. Isoparametric unfitted FEM
  12. Isoparametric finite element spaces
  13. Stabilized method for unique continuation
  14. Norms
  15. Stability, consistency and continuity
  16. ...and 15 more sections

Key Result

theorem 1

Let $u \in H^{q+1}(\Omega_{1} \cup \Omega_{2}) \cap W^{3,\infty}(\Omega_{1} \cup \Omega_{2})$ be the exact solution of (eq:PDE_bulk)-(eq:measurements). Let $(u_h,z_h) \in V_{h,\Theta} ^{\Gamma} \times V_{h,\Theta} ^{0}$ be the solution of (eq:discr_stab_var_form). Assume that $\omega \subset B \su

Figures (9)

  • Figure 1: Sketch of geometry and data for unique continuation over an interface.
  • Figure 2: Domain of definition of stabilization terms on the piecewise linear reference geometry ($q=1$). Facets in $\mathcal{F}^1$ are indicated by red dashed lines.
  • Figure 3: Figure (A) shows the reference solution (\ref{['eq:refsol-diffusion']}) in the purely diffusive case as a function of $\left\lVert x\right\rVert_4$ for different levels of the contrast. Figure (B) and (C) display a sketch of the geometrical setup for the experiments of Section \ref{['ssection:numexp-pure-diffusion']}, respectively \ref{['sssection:Helmholtz-contrast']}.
  • Figure 4: Dependence of the relative $L^2$-error in $B$ on the contrast for a pure diffusion problem in the geometry shown in Figure \ref{['fig:squares']}. In the insets the absolute error for the cases indicated by the arrows is shown. Here we used $q=p$. On the $x$-axis the number of degrees of freedom 'ndof' is displayed.
  • Figure 5: Dependence of the relative $L^2$-error in $B$ in terms of the choice of stabilization parameters. We consider a pure diffusion problem with $(\mu_1,\mu_2) = (20,2)$ in the geometry shown in Figure \ref{['fig:squares']}. The solid lines in the left plot show the results for the method as defined and analyzed in this paper, while dashed lines display the results for an alternative method in which the term $N_h^c$ in the bilinear form is omitted.
  • ...and 4 more figures

Theorems & Definitions (15)

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  • ...and 5 more