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ACMS Abstracts: Spring 2015

Irene Kyza (U Dundee)

Adaptivity and blowup detection for semilinear evolution convection-diffusion equations based on a posteriori error control

We discuss recent results on the a posteriori error control and adaptivity for an evolution semilinear convection-diffusion model problem with possible blowup in finite time. This belongs to the broad class of partial differential equations describing e.g., tumor growth,chemotaxis and cell modelling. In particular, we derive a posteriori error estimates that are conditional (estimates which are valid under conditions of a posteriori ­type) for an interior penalty discontinuous Galerkin (dG) implicit-explicit (IMEX) method using a continuation argument. Compared to a previous work, the obtained conditions are more localised and allow the efficient error control near the blowup time. Utilising the conditional a posteriori estimator we are able to propose an adaptive algorithm that appears to perform satisfactorily. In particular, it leads to good approximation of the blowup time and of the exact solution close to the blowup. Numerical experiments illustrate and complement our theoretical results. This is joint work with A. Cangiani, E.H. Georgoulis, and S. Metcalfe from the University of Leicester.

Tao Zhou (Chinese Academy of Sciences)

The Christoffel function weighted least-squares for stochastic collocation approximations: applications to Uncertainty Quantification

We shall consider the multivariate stochastic collocation methods on unstructured grids. The motivation for such a study is the applications in parametric Uncertainty Quantification (UQ). We will first give a general framework of stochastic collocation methods, which include approaches such as compressed sensing, least-squares, and interpolation. Particular attention will be then given to the least-squares approach, and we will review recent progresses in this topic.