# Welcome to modepy’s documentation!¶

modepy helps you create well-behaved high-order discretizations on simplices (i.e. triangles and tetrahedra). These are a key building block for high-order unstructured discretizations, as often used in a finite element context. It closely follows 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

The basic objects that modepy manipulates are functions on a simplex. For example, it supplies an orthonormal basis on triangles (shown here) and tetrahedra.

The file that created this plot is included in the modepy distribution as examples/plot-basis.py.

Here’s an idea of code that uses modepy:

import numpy as np
import modepy as mp

n = 17 # use this total degree
dimensions = 2

# Get a basis of orthonormal functions, and their derivatives.

basis = mp.simplex_onb(dimensions, n)

nodes = mp.warp_and_blend_nodes(dimensions, n)
x, y = nodes

# We want to compute the x derivative of this function:

f = np.sin(5*x+y)
df_dx = 5*np.cos(5*x+y)

# The (generalized) Vandermonde matrix transforms coefficients into
# nodal values. So we can find basis coefficients by applying its
# inverse:

f_coefficients = np.linalg.solve(
mp.vandermonde(basis, nodes), f)

# Now linearly combine the (x-)derivatives of the basis using
# f_coefficients to compute the numerical derivatives.

df_dx_num = np.dot(