kvadratisen3

Kvadratisen3 is a theoretical framework in numerical analysis that describes a family of three-dimensional quadrature schemes intended for exact integration of polynomials up to a specified degree on hexahedral elements. The term appears in discussions of high-order numerical integration and is not tied to a single canonical rule. The naming convention signals a progression of rules in a hypothetical sequence of methods.

Each kvadratisen3 rule consists of a finite set of integration nodes in the reference cube [-1,1]^3 and corresponding weights. The nodes can include tensor-product Gauss-Legendre points and additional off-axis points chosen so that the resulting quadrature is exact for all monomials up to a target degree. The rules exploit symmetry to minimize the number of distinct weights and are typically constructed by solving systems of moment equations.

Variants of kvadratisen3 differ in degree of exactness and node count. Design goals include maximizing the degree for a given number of nodes or minimizing nodes for a target degree. The rules are invariant under rigid body transformations of the element and are compatible with standard isoparametric mappings used in finite element methods.

Applications include finite element analysis, spectral element methods, and computational physics for accurate volume integrals, especially in high-order elements and p-refinement scenarios. Limitations involve potentially high node counts and increased assembly cost, which can outweigh benefits for low-order elements or curved geometries. In practice, kvadratisen3 rules are often considered alongside alternative quadrature strategies, such as reduced or adaptive schemes.

See also: quadrature rules, finite element method, numerical integration.