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The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611\u2013635] if this method was redesigned to make use of trial functions that satisfy a given problem\u2019s boundary conditions. A link between the new Petrov\u2013Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.<\/jats:p>","DOI":"10.1137\/22m1492842","type":"journal-article","created":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T14:22:45Z","timestamp":1675434165000},"page":"A123-A146","source":"Crossref","is-referenced-by-count":3,"title":["A Generic and Strictly Banded Spectral Petrov\u2013Galerkin Method for Differential Equations with Polynomial Coefficients"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3293-7573","authenticated-orcid":true,"given":"Mikael","family":"Mortensen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Division of Mechanics, University of Oslo, Oslo 0851, Norway."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,2,7]]},"reference":[{"key":"ref1","unstructured":"List of Finite Element Software Packages, https:\/\/en.wikipedia.org\/wiki\/List_of_finite_element_software_packages."},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6504"},{"key":"ref3","volume-title":"Chebyshev and Fourier Spectral Methods","author":"Boyd J. 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