Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids (Q1387893)

The authors discuss two important elements in the high-order accurate spatial discretization over given arbitrary grids: the integration of the basis functions over arbitrary domains and quadrature approximations to general integrals over such domains. They obtain formulas for exact integrals of polynomials and quadrature approximations up to degree five for the line segment, triangle and tetrahedron. These three shapes are the building blocks for integration over arbitrary polyhedral or polygonal grids. A table of coefficients for quadrature symmetry groups and integrals for all three shapes is given, enabling one to construct symmetric quadrature formulas in a rational manner. A new method of the authors for obtaining the solutions in higher dimensions is no more complex than the one we are familiar with for dimension one.