{
  "abstract": "Collocation methods based on Birkhoff interpolation at Gaussian-type points are well-conditioned for one-dimensional initial and boundary value problems [Wang et al., {\\em SIAM J. Sci. Comput.} 36 (2014)], but the underlying construction does not extend directly to multiple dimensions. We address the long-standing ill-conditioning of multidimensional collocation methods for second-order elliptic-type problems. The key observation is that the second-order differentiation matrix and its inverse, the pseudospectral integration matrix (PSIM) constructed from Birkhoff interpolation at Legendre-Gauss-Lobatto points, are both similar to symmetric negative definite matrices. This enables stable diagonalisation of the dense, non-symmetric and ill-conditioned differentiation and integration matrices, even for thousands of collocation points, and leads to efficient multidimensional Birkhoff preconditioners. For variable-coefficient problems, the coefficients are incorporated directly into the diagonalisation and preconditioner construction, which is essential for highly anisotropic, high-contrast, oscillatory and degenerate elliptic operators. We provide spectral analysis and extensive two- and three-dimensional numerical experiments, demonstrating substantial reductions in condition numbers and nearly polynomial-degree-independent GMRES convergence while retaining high-order accuracy. The resulting Birkhoff-collocation schemes make multidimensional spectral collocation methods practical for challenging elliptic problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39353",
  "author_total": 4,
  "authors": [
    {
      "name": "Shunchang Li"
    },
    {
      "name": "Zixuan Gao"
    },
    {
      "name": "Yujian Jiao"
    },
    {
      "name": "Li-Lian Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fg93sejk6fvh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39353",
  "fulltext_url": null,
  "id": "4fg93sejk6fvh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39353",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7215028582",
  "pdf_url": "https://arxiv.org/pdf/2609.39353",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fg93sejk6fvh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39353",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39353"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions",
  "title_text": "Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/4fg93sejk6fvh",
  "venue": "arXiv",
  "versions": []
}