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Speaker: Thomas Führer, Universidad Católica de Chile
Title: How much information can we recover from piecewise polynomial approximations?
Abstract: In this talk I present some recent results where we show how to define higher-order polynomial approximations from lower-order piecewise polynomial approximations. Under certain assumptions, which are met for solutions of many finite element methods, we prove that the novel approximation converges at higher rates. We present these ideas in the framework of quasi-interpolations. These are defined by taking a weighted average of the orthogonal projection onto piecewise polynomials. We prove that the novel operators enjoy optimal approximation properties. This can be exploited to define postprocessed solutions of finite element methods such as (hybridizable) discontinuous Galerkin methods. While popular postprocessing techniques achieve the same order of convergence and can be computed more efficiently, our approach extends to more general situations where, e.g., discrete gradient approximations are not directly accessible. Throughout the talk we present numerical examples. Our main results are based on technical results that analyze the intersection of orthogonal polynomials on patches.
This is a joint work with Manuel A. Sanchez.