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modepy: Basis Functions, Node Sets, Quadratures

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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
  • 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

Hesthaven, Jan S., and Tim Warburton. "Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications". 1st ed. Springer, 2007. Book web page

but much has been added beyond that basic functionality.

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Modes and nodes for high-order discretizations

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