A study of isogeometric analysis for scalar convection-diffusion equations

Isogeometric Analysis (IGA), in combination with the Streamline Upwind Petrov--Galerkin (SUPG) stabilization, is studied for the discretization of steady-state con\-vection-diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers

Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

Stabilized Isogeometric Collocation Methods For Scalar Transport and Incompressible Fluid Flow

In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation schemes address spurious oscillations that can arise from advection and pressure instabilities. Numerical examples for the advection-diffusion equation, Stokes equations, and incompressible Navier-Stokes equations show the effectiveness of the proposed stabilized schemes while maintaining the high-order convergence rates and accuracy of standard isogeometric collocation on smooth problems

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield