A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are i) non-repeating interior point operators, ii) nonuniform nodal distribution in the computational domain, iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditio...

Gauss quadrature rules associated with a nonnegative measure support on (part of) the real axis find many applications in Scientific Computing. It is important to be able estimate error when replacing an integral by ℓ-node rule order choose suitable number of nodes. A classical approach this evaluate (2ℓ+1)-node Gauss–Kronrod rule. However, 2ℓ+1 nodes might not exist. The generalized averaged f...

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε−1 and d. Strong tractability means that ...

We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the “half-point rule”, indicating that optimal rules involve a number of points roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order o...