  In this work, we use the monolithic convex limiting (MCL) methodology to
enforce relevant inequality constraints in implicit finite element
discretizations of the compressible Euler equations. In this context,
preservation of invariant domains follows from positivity preservation for
intermediate states of the density and internal energy. To avoid spurious
oscillations, we additionally impose local maximum principles on intermediate
states of the density, velocity components, and specific total energy. For the
backward Euler time stepping, we show the invariant domain preserving (IDP)
property of the fully discrete MCL scheme by constructing a fixed-point
iteration that is IDP and converges under a strong time step restriction. Our
iterative solver for the nonlinear discrete problem employs a more efficient
fixed-point iteration. The matrix of the associated linear system is a robust
low-order Jacobian approximation that exploits the homogeneity property of the
flux function. The limited antidiffusive terms are treated explicitly. We use
positivity preservation as a stopping criterion for nonlinear iterations. The
first iteration yields the solution of a linearized semi-implicit problem. This
solution possesses the discrete conservation property but is generally not IDP.
Further iterations are performed if any non-IDP states are detected. The
existence of an IDP limit is guaranteed by our analysis. To facilitate
convergence to steady-state solutions, we perform adaptive explicit
underrelaxation at the end of each time step. The calculation of appropriate
relaxation factors is based on an approximate minimization of nodal entropy
residuals. The performance of proposed algorithms and alternative solution
strategies is illustrated by the convergence history for standard
two-dimensional test problems.
