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Spectral pollution and second order relative spectra for self-adjoint operators

Michael Levitin, Eugene Shargorodsky

TL;DR

This work addresses spectral pollution that arises when approximating spectra of self-adjoint operators via projection methods. It introduces the second order relative spectrum $Spec_2(A, \mathcal{L})$ as an a posteriori detector that locates genuine spectral points when approximate spectra converge, providing a quantitative link: $z\\in Spec_2(A, \mathcal{L})$ implies $Spec(A)\\cap [\\Re z - |\\Im z|, \\Re z + |\\Im z|] \\neq \\emptyset$. Theoretical results are complemented by two model problems: a discontinuous multiplication operator and a Stokes-type ODE system, where standard projection methods exhibit pollution that is effectively identified and filtered by $Spec_2$. The findings demonstrate that carefully chosen mixed-order finite element spaces, together with second order relative spectra, can yield accurate spectra and robust pollution detection, suggesting practical applicability to complex operators beyond the toy models.

Abstract

We consider the phenomenon of spectral pollution arising in calculation of spectra of self-adjoint operators by projection methods. We suggest a strategy of dealing with spectral pollution by using the so-called second order relative spectra. The effectiveness of the method is illustrated by a detailed analysis of two model examples.

Spectral pollution and second order relative spectra for self-adjoint operators

TL;DR

This work addresses spectral pollution that arises when approximating spectra of self-adjoint operators via projection methods. It introduces the second order relative spectrum as an a posteriori detector that locates genuine spectral points when approximate spectra converge, providing a quantitative link: implies . Theoretical results are complemented by two model problems: a discontinuous multiplication operator and a Stokes-type ODE system, where standard projection methods exhibit pollution that is effectively identified and filtered by . The findings demonstrate that carefully chosen mixed-order finite element spaces, together with second order relative spectra, can yield accurate spectra and robust pollution detection, suggesting practical applicability to complex operators beyond the toy models.

Abstract

We consider the phenomenon of spectral pollution arising in calculation of spectra of self-adjoint operators by projection methods. We suggest a strategy of dealing with spectral pollution by using the so-called second order relative spectra. The effectiveness of the method is illustrated by a detailed analysis of two model examples.

Paper Structure

This paper contains 8 sections, 4 theorems, 53 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Example I: operator of multiplication by a discontinuous function
  4. Example II: Stokes type system of ODEs
  5. Proofs of the main results
  6. More on Example I
  7. More on Example II
  8. Concluding remarks

Key Result

Theorem 2.1

For any $\lambda \in \textup{conv}\left(\mathop{\mathrm{\widehat{S}pec}}\nolimits_{\textup{ess}}(A)\right)\setminus \mathop{\mathrm{\widehat{S}pec}}\nolimits_{\textup{ess}}(A)$ there exists an increasing sequence $(\mathcal{L}_k)_{k \in \mathbb{N}} \in \Lambda(A)$ such that

Figures (18)

  • Figure 1: $\mathop{\mathrm{Spec}}\nolimits(A) \cap [\mathop{\mathrm{Re}}\nolimits z - |\mathop{\mathrm{Im}}\nolimits z|,\mathop{\mathrm{Re}}\nolimits z + |\mathop{\mathrm{Im}}\nolimits z|]\not= \varnothing$.
  • Figure 2: $\mathop{\mathrm{Spec}}\nolimits(A,\mathcal{L}_{1001})$.
  • Figure 3: A typical computed "eigenfunction" of $A$ corresponding to a spurious eigenvalue.
  • Figure 4: $\mathop{\mathrm{Spec}}\nolimits_2(A, \mathcal{L}_N)$.
  • Figure 5: Detailed view of Figure \ref{['fig:spec2_A']} near the real axis.
  • ...and 13 more figures

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 4.1
  • proof : Proof of Theorem \ref{['poll']}
  • Remark 5.1
  • ...and 6 more