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In many applications the accurate representation of the computational domainis a key factor to obtain reliable and effective numerical solutions. Curvedinterfaces, which might be internal, related to physical data, or portions ofthe physical boundary, are often met in real applications. However, they areoften approximated leading to a geometrical error that might become dominantand deteriorate the quality of the results. Underground problems often involvethe motion of fluids where the fundamental governing equation is the Darcy law.High quality velocity fields are of paramount importance for the successfulsubsequent coupling with other physical phenomena such as transport. Thevirtual element method, as solution scheme, is known to be applicable inproblems whose discretizations requires cells of general shape, and the mixedformulation is here preferred to obtain accurate velocity fields. To overcomethe issues associated to the complex geometries and, at the same time,retaining the quality of the solutions, we present here the virtual elementmethod to solve the Darcy problem, in mixed form, in presence of curvedinterfaces in two and three dimensions. The numerical scheme is presented indetail explaining the discrete setting with a focus on the treatment of curvedinterfaces. Examples, inspired from industrial applications, are presentedshowing the validity of the proposed approach.

The mixed virtual element method for grids with curved interfaces