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: