modepy 2026.1

Modes and nodes for high-order methods

modepy helps you create well-behaved high-order discretizations on simplices (i.e. segments, triangles and tetrahedra) and tensor products of simplices (i.e. squares, cubes, prisms, etc.). These are a key building block for high-order unstructured discretizations, as often used in a finite element context. Features include:

• Support for simplex and tensor product elements in any dimension.

• Orthogonal bases:

  • Jacobi polynomials with derivatives

  • Orthogonal polynomials for simplices up to 3D and tensor product elements and their derivatives.

  • All bases permit symbolic evaluation, for code generation.

• Access to numerous quadrature rules:

  • Jacobi-Gauss, Jacobi-Gauss-Lobatto in 1D (includes Legendre, Chebyshev, ultraspherical, Gegenbauer)

  • Clenshaw-Curtis and Fejér in 1D

  • Grundmann-Möller on the simplex

  • Xiao-Gimbutas on the simplex

  • Vioreanu-Rokhlin on the simplex

  • Jaśkowiec-Sukumar on the tetrahedron

  • Witherden-Vincent on the hypercube

  • Generic tensor products built on the above, e.g. for prisms and hypercubes

• Tools to construct new quadrature rules:

  • A basic iterative Gauss-Newton process to optimize/tighten rules

  • Vioreanu-Rokhlin node initial generation based on multiplication operators

• Matrices for FEM, usable across all element types:

  • generalized Vandermonde,

  • mass matrices (including lumped diagonal),

  • face mass matrices,

  • differentiation matrices, and

  • resampling matrices.

• Objects to represent ‘element shape’ and ‘function space’, generic node/mode/quadrature retrieval based on them.

Its roots closely followed the approach taken in the book

J. S. Hesthaven and T. Warburton (2007). Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (1st ed.). doi:10.1007/978-0-387-72067-8. (source code)

but much has been added beyond that basic functionality.

Resources:

• Documentation.

• Source code.

• Bug tracker.

• PyPI.