The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible
fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of
capillarity effects close to phase boundaries. Standard numerical discretizations are
known to violate discrete versions of inherent energy inequalities, thus leading to
spurious dynamics of computed solutions close to static equilibria (e.g.,
parasitic currents). In this work, we propose a time-implicit discretization of the
problem, and use piecewise linear (or bilinear), globally continuous finite element spaces
for both, velocity and density fields, and two regularizing terms where corresponding
parameters tend to zero as the mesh-size h > 0 tends to zero.
Solvability, non-negativity of computed densities, as well as conservation of mass, and a
discrete energy law to control dynamics are shown. Computational experiments are provided
to study interesting regimes of coefficients for viscosity and capillarity.