Quasi-interpolators with application to post-processing in finite-element methods

Abstract 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 $p+1$. The operators deal with both quasi-interpolation and post-processing. We show that the operators have optimal approximation properties in the $L^{2}$ norm, i.e., of order $p+2$. This can be exploited to enhance the accuracy of finite-element approximations provided 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 $p$. Various numerical schemes, such as mixed finite-element methods and discontinuous Petrov–Galerkin methods, meet this condition. Contrary to well-established post-processing techniques that also require this or a similar closeness property, our proposed method delivers a conforming post-processed 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 post-process 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.