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This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non-uniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in Kim et al. (2013), we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal L 2 error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for L 2 error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented. • The method is robust in both Stokes and Darcy limits, that is, the error of approximation is uniform with respect to the parameters in the equations. • A new idea to enforce continuity of numerical solutions across cell boundaries is developed. • The optimal convergence of the method is proved. Also, a superconvergence result is proved. • The method is very general, and it can be used with very general polygonal meshes. [ABSTRACT FROM AUTHOR]