Raviart--Thomas Elements with Geometric Correction for Distorted Quadrilateral Meshes

Mixed finite element methods based on Raviart--Thomas spaces are widely used for the numerical approximation of second--order elliptic problems in flux form. On quadrilateral meshes, however, the bilinear mapping from the reference element introduces a spatially varying Jacobian, which may violate the inclusion property $\mathrm{div}\,V_h \subset W_h$ for the standard Raviart--Thomas spaces.
In this paper we propose a simple modification of the classical Raviart--Thomas elements on quadrilateral meshes. The modification consists of adding geometrically motivated correction terms to the local basis functions in order to compensate for the geometric distortion introduced by the bilinear mapping. The resulting spaces retain the same dimension and degrees of freedom as the classical Raviart--Thomas elements while restoring the compatibility property.
We present a general framework for constructing such modified spaces and illustrate the approach by developing modified versions of the lowest order and next--to--lowest order Raviart--Thomas elements. Theoretical analysis establishes optimal approximation properties under the standard shape--regularity assumption for quadrilateral meshes. Numerical experiments on distorted meshes confirm the predicted convergence rates and show that the modified elements yield consistently improved accuracy over the classical Raviart--Thomas formulation as the geometric distortion increases.