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.

翻译：基于次数不超过$p$的分片多项式权函数，我们构造了映射到次数不超过$p+1$的连续分片多项式空间的拟插值算子。证明了这些算子具有最优逼近性质，即达到$p+2$阶精度。当有限元解充分接近精确解在次数不超过$p$的分片多项式空间上的正交投影时，该性质可用于提升有限元近似精度。混合有限元方法、不连续Petrov-Galerkin方法等多种数值格式均满足该条件。与同样需要此（或类似）逼近性质且依赖离散导数近似或底层偏微分方程局部版本的经典后处理技术不同，本方法直接提供满足相容性的后处理解。此外，我们引入基于分片常值权函数的第二类拟插值算子，可用于混合化间断Galerkin方法的解后处理。所提出算子的另一应用在于定义具有负指数Sobolev空间有界性的投影算子。数值算例验证了该方法的有效性。