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Analysis of FEAST spectral approximations using the DPG discretization

Jay Gopalakrishnan, Luka Grubišić, Jeffrey Ovall, Benjamin Q. Parker

TL;DR

This work develops a rigorous framework for FEAST-like spectral approximations of unbounded PDE operators using a DPG discretization of the resolvent. It proves absence of spectral pollution and derives a priori error bounds for the computed eigenspace and Ritz values, including explicit convergence rates that depend on elliptic regularity and discretization degree. The theory is validated numerically on unit-square and L-shaped domains, and demonstrated on a realistic optical-fiber problem to compute guided modes; results show high-order convergence and practical applicability. The combination of FEAST with DPG enables efficient, accurate computation of eigenvalue clusters in PDE settings and provides a foundation for robust eigenspace control and adaptive strategies in engineering applications.

Abstract

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as "FEAST", has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

Analysis of FEAST spectral approximations using the DPG discretization

TL;DR

This work develops a rigorous framework for FEAST-like spectral approximations of unbounded PDE operators using a DPG discretization of the resolvent. It proves absence of spectral pollution and derives a priori error bounds for the computed eigenspace and Ritz values, including explicit convergence rates that depend on elliptic regularity and discretization degree. The theory is validated numerically on unit-square and L-shaped domains, and demonstrated on a realistic optical-fiber problem to compute guided modes; results show high-order convergence and practical applicability. The combination of FEAST with DPG enables efficient, accurate computation of eigenvalue clusters in PDE settings and provides a foundation for robust eigenspace control and adaptive strategies in engineering applications.

Abstract

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as "FEAST", has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

Paper Structure

This paper contains 11 sections, 6 theorems, 72 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The abstract framework
  3. Application to a DPG discretization
  4. The Dirichlet operator
  5. The DPG resolvent discretization
  6. FEAST iterations with the DPG discretization
  7. A generalization to additive perturbations
  8. Numerical convergence studies
  9. Discretization errors on the unit square
  10. Convergence rates on an L-shaped domain
  11. Application to optical fibers

Key Result

Theorem 2.1

Suppose Assumptions asm:rN--asm:Rlim hold. Then there are constants $C_N, h_0>0$ such that, for all $h < h_0$, If, in addition, Assumption asm:Vh_in_dom(a) holds and $\| u \|_{\mathcal{V}} = \| |A|^{1/2} u \|_{\mathcal{H}}$, then there are $C_1, h_1>0$ such that for all $h < h_1,$ where $\Lambda_h^{\mathrm{max}} = \sup_{e_h \in E_h} \| |A|^{1/2} e_h\|_{{\mathcal{H}}} / \| e_h \|_{\mathcal{H}}$ sa

Figures (3)

  • Figure 1: Results for the unit square
  • Figure 2: The mesh used for computing modes of the ytterbium-doped fiber.
  • Figure 3: A close view of the approximate eigenfunctions $\varphi_j^h$ computed by FEAST for the ytterbium-doped fiber. The boundary of the fiber core region is marked by dashed black circles.

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 3 more