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# The ordering of Basix's polynomial sets

When defining finite elements, Basix uses orthonormal polyomials to represent a basis of the space spanned by the finite element functions. This page details the order in which these polynomials are stored.

In each example below, we give a list of the monomials that appear for the first time in each term. The actual orthogonal polynomials will be linear combinations of the monomial and all the previous monomials. For example, the monomials

\(1\), \(x\), \(x^2\), \(x^3\), ...

are given below for an interval. The orthonormal polynomials on an interval, however, are

\(1\), \(\sqrt{3}(2x-1)\), \(\sqrt{5}(6x^2-6x+1)\), \(\sqrt{6}(20x^3 - 30x^2 + 12x - 1)\), ...

### Interval

On an interval, Basix orders its polynomials by lowest degree first, ie

\(1\), \(x\), \(x^2\), \(x^3\), ...

### Triangle

On a triangle, Basix orders its polynomials by degree (lowest first). Polynomials of the same degree are ordered so that higher powers of \(y\) appear first, ie

\(1\), \(y\), \(x\), \(y^2\), \(xy\), \(x^2\), \(y^3\), \(x^2y\), \(xy^2\), \(x^3\), ...

### Tetrahedron

On a tetrahedron, Basix orders its polynomials by degree (lowest first). Polynomials of the same degree are ordered so that higher powers of \(z\) appear first then higher powers of \(y\), ie

\(1\), \(z\), \(y\), \(x\), \(z^2\), \(yz\), \(xz\), \(y^2\), \(xy\), \(x^2\), \(z^3\), \(yz^2\), \(xz^2\), \(y^2z\), \(xyz\), \(x^2z\), \(y^3\), \(xy^2\), \(x^2y\), \(x^3\), ...

### Quadrilateral

On a quadrilateral, we take a tensor product of the polynomials on an interval with \(y\) as the variable and those with \(x\) as the variable. The resulting ordering has polynomials containing only \(y\) first (low-to-high degree), then these polynomials times \(x\), then the same polynomials times \(x^2\), and so on; ie

\(1\), \(y\), \(y^2\), \(y^3\), \(x\), \(xy\), \(xy^2\), \(xy^3\), \(x^2\), \(x^2y\), \(x^2y^2\), \(x^2y^3\), \(x^3\), \(x^3y\), \(x^3y^2\), \(x^3y^3\)

### Hexahedron

On a hexahedron, we take a tensor product of the polynomials on an interval with \(z\) as the variables and the polynomials on a quadrilateral with \((x,y)\) as the variables. The resulting ordering has polynomials containing only \(z\) first (low-to-high degree) then these polynomials times \(y\), then the same polynomials times \(y^2\), then \(y^3\) and so on (following the order on a quadrilateral); ie

\(1\), \(z\), \(z^2\), \(z^3\), \(y\), \(yz\), \(yz^2\), \(yz^3\), \(y^2\), \(y^2z\), \(y^2z^2\), \(y^2z^3\), \(y^3\), \(y^3z\), \(y^3z^2\), \(y^3z^3\), \(x\), \(xz\), \(xz^2\), \(xz^3\), \(xy\), \(xyz\), \(xyz^2\), \(xyz^3\), \(xy^2\), \(xy^2z\), \(xy^2z^2\), \(xy^2z^3\), \(xy^3\), \(xy^3z\), \(xy^3z^2\), \(xy^3z^3\), \(x^2\), \(x^2z\), \(x^2z^2\), \(x^2z^3\), \(x^2y\), \(x^2yz\), \(x^2yz^2\), \(x^2yz^3\), \(x^2y^2\), \(x^2y^2z\), \(x^2y^2z^2\), \(x^2y^2z^3\), \(x^2y^3\), \(x^2y^3z\), \(x^2y^3z^2\), \(x^2y^3z^3\), \(x^3\), \(x^3z\), \(x^3z^2\), \(x^3z^3\), \(x^3y\), \(x^3yz\), \(x^3yz^2\), \(x^3yz^3\), \(x^3y^2\), \(x^3y^2z\), \(x^3y^2z^2\), \(x^3y^2z^3\), \(x^3y^3\) \(x^3y^3z\), \(x^3y^3z^2\), \(x^3y^3z^3\)

### Prism

On a hexahedron, we take a tensor product of the polynomials on an interval with \(z\) as the variables and the polynomials on a triangle with \((x,y)\) as the variables. The resulting ordering has polynomials containing only \(z\) first (low-to-high degree) then these polynomials times \(y\), then the same polynomials times \(x\), then \(y^2\) and so on (following the order on a triangle); ie

\(1\), \(z\), \(z^2\), \(y\), \(yz\), \(yz^2\), \(x\), \(xz\), \(xz^2\), \(y^2\), \(y^2z\), \(y^2z^2\), \(xy\), \(xyz\), \(xyz^2\), \(x^2\), \(x^2z\), \(x^2z^2\), ...