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Quasi-Interpolation Operators for Enhancing Finite Element Approximations


Core Concepts
The authors design quasi-interpolation operators that can be used to enhance the accuracy of finite element approximations, particularly for mixed finite element methods and discontinuous Petrov-Galerkin methods, without requiring the use of discrete approximations of derivatives or local versions of the underlying PDE.
Abstract
The authors introduce two families of quasi-interpolation operators: The first family, Jp+1 0, uses piecewise polynomial weight functions of degree up to p. These operators have optimal approximation properties, i.e., of order p+2, provided that the finite element solution is sufficiently close to the orthogonal projection of the exact solution onto the space of piecewise polynomials of degree up to p. This property is satisfied by various numerical schemes, such as mixed finite element methods and discontinuous Petrov-Galerkin methods. The second family, Ip+1 0, uses piecewise constant weight functions. These operators have similar properties as the first family, but the existence of the weight functions requires certain assumptions on the underlying mesh. The authors also demonstrate that the quasi-interpolators can be used to define projection operators onto the space of piecewise polynomials of degree up to p that are uniformly bounded in negative order Sobolev spaces. Numerical examples are provided to showcase the effectiveness of the proposed quasi-interpolation operators.
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Deeper Inquiries

How can the proposed quasi-interpolation operators be extended to higher-dimensional problems or more general mesh structures

The proposed quasi-interpolation operators can be extended to higher-dimensional problems or more general mesh structures by adapting the construction of the weight functions and the definition of the operators. In the context of higher-dimensional problems, the weight functions can be defined over higher-dimensional patches or neighborhoods around the nodes, edges, or elements. This extension would involve considering the geometry and connectivity of the elements in higher dimensions to ensure the weight functions capture the local information accurately. Additionally, the operators can be formulated to account for the increased complexity of the mesh structures in higher dimensions, such as tetrahedral or hexahedral elements, by appropriately defining the basis functions and constraints for the weight functions.

What are the potential limitations or drawbacks of the quasi-interpolation approach compared to other postprocessing techniques, such as those based on discrete approximations of derivatives or local versions of the underlying PDE

While quasi-interpolation offers advantages in terms of preserving polynomial properties and providing conforming postprocessed solutions without relying on discrete approximations of derivatives or local PDE approximations, there are potential limitations and drawbacks to consider. One limitation is the requirement for specific conditions on the mesh, such as the existence of certain neighborhoods or patches around the nodes, edges, or elements, which may not always be feasible or straightforward to satisfy in practical applications. Additionally, the construction and computation of the weight functions for quasi-interpolation can be computationally intensive, especially for higher-dimensional problems or complex mesh structures, which may impact the efficiency of the postprocessing approach. Furthermore, the optimal approximation properties of quasi-interpolation may be limited to certain classes of problems or numerical schemes, and the effectiveness of the method could vary depending on the characteristics of the underlying finite element approximation.

Could the quasi-interpolation operators be combined with adaptive mesh refinement strategies to further enhance the accuracy of finite element approximations

The quasi-interpolation operators can be effectively combined with adaptive mesh refinement strategies to further enhance the accuracy of finite element approximations. By incorporating quasi-interpolation into adaptive mesh refinement algorithms, the postprocessing step can adaptively adjust the mesh resolution based on the error estimates provided by the quasi-interpolation operators. This adaptive approach allows for targeted refinement in regions where the approximation error is high, leading to improved accuracy and efficiency in the finite element solution. Additionally, the combination of quasi-interpolation with adaptive mesh refinement can help optimize the computational resources by focusing on refining the mesh only in areas where it is most beneficial for improving the solution accuracy. Overall, this integration of quasi-interpolation with adaptive mesh refinement strategies offers a powerful tool for enhancing the performance of finite element methods in a wide range of applications.
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