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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