Construction and Optimization of Summation-by-Parts Operators for General Function Spaces Using an Improved Generalized Gaussian Quadrature Algorithm

Jul 16, 2026

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We construct optimized summation-by-parts (SBP) operators for general function spaces on open, closed, and half-open nodal distributions, achieving provably minimal degrees of freedom. The operators are built from generalized Gaussian quadrature rules, supported by an improved algorithm that is flexible, efficient, and provably convergent. When free parameters are present, two optimization strategies further enhance accuracy. Numerical tests with large or unbounded gradients show orders-of-magnitude improvements over standard polynomial operators and over equispaced nodal function-space SBP methods. Operator optimization is critical to avoid nullspace inconsistency and poor conditioning.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

Introduction

General function-space SBP motivation

Challenges with uniformly spaced nodes

Generalized Gaussian quadrature and minimal DOF

Role of operator optimization

arXiv :2607 .08934v1 [math .NA] 9 Jul 2026 Noname manuscript No. (will be inserted by the editor) Construction and Optimization of Summation-byParts Operators for General Function Spaces Using an Improved Generalized Gaussian Quadrature Algorithm Alex Bercik · Lisa Patrascu · David W. Zingg Received: date / Accepted: date Abstract We construct optimized summation-by-parts (SBP) operators for general function spaces with provably minimal degrees of freedom on open, closed, and half-open nodal distributions. These operators rely on generalized Gaussian quadrature rules, for which we present an improved algorithm that is flexible, efficient, and provably convergent. In cases where free parameters are available, we further introduce two operator optimization strategies. We test our operators on a handful of numerical examples that contain large or unbounded gradients, in which some a priori knowledge of the solution has been assumed to select an appropriate basis. The novel operators are found to outperform standard polynomial operators by several orders of magnitude in solution accuracy relative to degrees of freedom. Furthermore, our novel operators significantly outperform function-space SBP operators with equispaced nodal distributions, which require significantly more nodes for the same operator basis. Finally, we demonstrate that the operator optimization procedures are critical to achieving accurate and efficient discretizations, as the standard SBP construction procedure can lead to nullspace-inconsistent and poorly-conditioned operators. Keywords Summation-by-parts · High-order methods · Generalized Gauss Quadrature · General function spaces Mathematics Subject Classification (2020) 65D25 · 65D32 · 65M06 · 65M60 · 65M70 A. Bercik University of Toronto Institute for Aerospace Studies, Toronto, Canada E-mail: [alex.bercik@mail.utoronto.ca](alex.bercik@mail.utoronto.ca) L. Patrascu University of Toronto Institute for Aerospace Studies, Toronto, Canada E-mail: [lisa.patrascu@mail.utoronto.ca](lisa.patrascu@mail.utoronto.ca) [D. W. Zingg](D. W. Zingg) University of Toronto Institute for Aerospace Studies, Toronto, Canada E-mail: [david.zingg@utoronto.ca](david.zingg@utoronto.ca) 1 Introduction Summation-by-parts (SBP) operators, together with simultaneous approximation terms (SATs) for enforcing boundary conditions, allow for the construction of provably robust high-order discretizations of partial differential equations. By satisfying integration by parts at the semidiscrete level, they provide an algebraic framework for reproducing stability and conservation proofs of the continuous problem [29, 39 , 9 , 11 , 40] . As with any numerical method that achieves high-order through polynomial approximation, however, accuracy can suffer when applied to practical problems of interest involving singularities, high gradients, or other features requiring high resolution, such as cracks in solid mechanics [8, 15 , 4], or boundary layers and singularities in fluid mechanics [44, 28 , 7 , 47 , 14] . The use of nonpolynomial basis functions to improve the accuracy of a numerical method without remeshing has been studied in the context of the finite element method as early as the 1970s [8, 15 , 4] . Several closely related methods have since emerged, including the Extended/Generalized Finite Element Methods (XFEM/GFEM) [3, 38 , 16], the Discontinuous Enrichment Method (DEM) [13], and the Discontinuous Galerkin (DG) method with non-polynomial approximation spaces [46] . The XFEM/GFEM can be classified under the broader definition of a partition of unity method [1, 36 , 2], where the basis is augmented by multiplications of nonpolynomial ‘enrichment functions’ with the partition of unity basis shape functions. This allows discontinuities and singularities tobe more effectively represented while maintaining the compact support of the local basis functions. For a comprehensive overview of these methods, we

FAQs

What problem do the paper’s optimized summation-by-parts (SBP) operators address?

They aim to build robust high-order SBP discretizations for general function spaces while minimizing degrees of freedom across open, closed, and half-open nodal distributions.

How are the new SBP operators constructed?

They rely on generalized Gaussian quadrature rules, using an improved quadrature algorithm that is flexible, efficient, and provably convergent.

Why are operator optimization procedures important in the reported results?

Without optimization, the standard SBP construction procedure can yield nullspace-inconsistent and poorly conditioned operators, harming accuracy and efficiency.

Jul 16, 2026

Upload by Riley