Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions
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.
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BibTeX
@misc{li2026wellconditioned,
title = {{Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions}},
author = {Shunchang Li and Zixuan Gao and Yujian Jiao and Li-Lian Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39353},
eprint = {2609.39353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39353},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-Conditioned Birkhoff-Collocation Methods for Elliptic-type Problems in Multiple Dimensions
AU - Shunchang Li
AU - Zixuan Gao
AU - Yujian Jiao
AU - Li-Lian Wang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39353
UR - https://arxiv.org/abs/2609.39353
AB - 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.
ER -