The mixed virtual element method for grids with curved interfaces

TL;DR

This paper introduces a mixed virtual element method tailored for accurately solving Darcy flow problems with curved interfaces in 2D and 3D, addressing geometric errors and improving velocity field quality in complex geometries.

Contribution

It develops a novel virtual element approach that effectively handles curved interfaces, enhancing solution accuracy in complex geometrical domains.

Findings

The method accurately captures flow in geometries with curved interfaces.

Numerical examples demonstrate improved velocity field quality.

The approach is applicable in industrial-like scenarios.

Abstract

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