@article {ipa|Adaptive isogeometric methods with hierarchical splines: error estimator and convergence|2015, title = {Adaptive isogeometric methods with hierarchical splines: error estimator and convergence}, year = {2015}, month = {January}, pages = {30 p.}, institution = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, abstract = {

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally rened meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.

}, keywords = {Numerical schemes}, url = {http://irs.imati.cnr.it/reports/irs15-01}, author = {A. Buffa and C. Giannelli} } @article {ipa|On quasi-interpolation operators in spline spaces|2015, title = {On quasi-interpolation operators in spline spaces}, year = {2015}, month = {December}, pages = {16 p.}, institution = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, abstract = {

We propose the construction of a class of L2 stable quasi-interpolation operators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot repetition and to knot "vicinity" (up to p+1 knots), so it applies to the most general scenario in which the B-spline functions are known to be well defined.

}, keywords = {Quasi-interpolation}, url = {http://irs.imati.cnr.it/reports/irs15-03}, author = {A. Buffa and E.M. Garau and C. Giannelli and G. Sangalli} }