8 research outputs found
Adaptive isogeometric analysis with hierarchical box splines
Isogeometric analysis is a recently developed framework based on finite
element analysis, where the simple building blocks in geometry and solution
space are replaced by more complex and geometrically-oriented compounds. Box
splines are an established tool to model complex geometry, and form an
intermediate approach between classical tensor-product B-splines and splines
over triangulations. Local refinement can be achieved by considering
hierarchically nested sequences of box spline spaces. Since box splines do not
offer special elements to impose boundary conditions for the numerical solution
of partial differential equations (PDEs), we discuss a weak treatment of such
boundary conditions. Along the domain boundary, an appropriate domain strip is
introduced to enforce the boundary conditions in a weak sense. The thickness of
the strip is adaptively defined in order to avoid unnecessary computations.
Numerical examples show the optimal convergence rate of box splines and their
hierarchical variants for the solution of PDEs
Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal