Quasi-interpolators with application to postprocessing in finite element methods

We design quasi-interpolation operators based on piecewise polynomial weight functions of degree less than or equal to $p$ that map into the space of continuous piecewise polynomials of degree less than or equal to $p+1$. We show that the operators have optimal approximation properties, i.e., of order $p+2$. This can be exploited to enhance the accuracy of finite element approximations provided that they are sufficiently close to the orthogonal projection of the exact solution on the space of piecewise polynomials of degree less than or equal to $p$. Such a condition is met by various numerical schemes, e.g., mixed finite element methods and discontinuous Petrov--Galerkin methods. Contrary to well-established postprocessing techniques which also require this or a similar closeness property, our proposed method delivers a conforming postprocessed solution that does not rely on discrete approximations of derivatives nor local versions of the underlying PDE. In addition, we introduce a second family of quasi-interpolation operators that are based on piecewise constant weight functions, which can be used, e.g., to postprocess solutions of hybridizable discontinuous Galerkin methods. Another application of our proposed operators is the definition of projection operators bounded in Sobolev spaces with negative indices. Numerical examples demonstrate the effectiveness of our approach.