  Internal interfaces in a domain could exist as a material defect or they can
appear due to propagations of cracks. Discretization of such geometries and
solution of the contact problem on the internal interfaces can be
computationally challenging. We employ an unfitted Finite Element (FE)
framework for the discretization of the domains and develop a tailored,
globally convergent, and efficient multigrid method for solving contact
problems on the internal interfaces. In the unfitted FE methods, structured
background meshes are used and only the underlying finite element space has to
be modified to incorporate the discontinuities. The non-penetration conditions
on the embedded interfaces of the domains are discretized using the method of
Lagrange multipliers. We reformulate the arising variational inequality problem
as a quadratic minimization problem with linear inequality constraints. Our
multigrid method can solve such problems by employing a tailored multilevel
hierarchy of the FE spaces and a novel approach for tackling the discretized
non-penetration conditions. We employ pseudo-$L^2$ projection-based transfer
operators to construct a hierarchy of nested FE spaces from the hierarchy of
non-nested meshes. The essential component of our multigrid method is a
technique that decouples the linear constraints using an orthogonal
transformation of the basis. The decoupled constraints are handled by a
modified variant of the projected Gauss-Seidel method, which we employ as a
smoother in the multigrid method. These components of the multigrid method
allow us to enforce linear constraints locally and ensure the global
convergence of our method. We will demonstrate the robustness, efficiency, and
level independent convergence property of the proposed method for Signorini's
problem and two-body contact problems.
