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Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Victor Dominguez, Catalin Turc

Abstract

Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of Boundary Integral Equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul-Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations which are typically encountered in the description of one dimensional closed curves.

Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Abstract

Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of Boundary Integral Equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul-Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations which are typically encountered in the description of one dimensional closed curves.
Paper Structure (18 sections, 13 theorems, 196 equations, 3 figures, 5 tables)

This paper contains 18 sections, 13 theorems, 196 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Navier equations and boundary element method
  3. Boundary value problem and Helmholtz decomposition
  4. Helmholtz BIOs
  5. Regularized BIEs with arc-length parametrizations
  6. Periodic Sobolev spaces and some useful operators
  7. Parameterized Boundary Integral Operators
  8. Arc-length parameterized regularized integral equations
  9. Nyström discretization for arc-length parametrizations BIEs and error analysis
  10. Nyström method
  11. Error analysis
  12. Arbitrary parametrizations
  13. The construction of the regularizing operators
  14. Parameterized Helmholtz BIO
  15. Nyström discretization
  16. ...and 3 more sections

Key Result

Lemma 3.1

It holds where, for $j=1{,}\ 3,\ 4$, $\mathop{\mathrm{V}}\nolimits_{k}^{(j)} \in\mathrm{OPS}(-j)$ are integral operators which can be written in terms of periodic integral operators with explicit kernels as where $A^{(j)}$ and $B^{(j)}$ are smooth bi-periodic functions.

Figures (3)

  • Figure 1: Geometries for the experiments considered in this section: The ellipse, the kite and the cavity domain. Notice that all the curves are of length $2\pi$.
  • Figure 2: Left panel: Eigenvalue distribution in the complex plane for the three geometries, with $N=1024$ and $\omega =10$, with natural parametrization. Right panel, a detail (zoom at) around the accumulation points, the eigenvalues of $\mathcal{H}_{p,s}$.
  • Figure 3: Left panel: Eigenvalue distribution in the complex plane for the three geometries, with $N=1024$ and $\omega =100$. Right panel, a detail (zoom at) around the accumulation points, the eigenvalues of $\mathcal{H}_{p,s}$

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Remark 3.6
  • Remark 4.1
  • Proposition 4.2
  • ...and 8 more