Weak and Strong Stabilisation of Cut Finite Element Methods

  • Burman, Erik (UCL)
  • Hansbo, Peter (Jönköping University)
  • Larson, Mats G (Umeå University)

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Cut finite element methods are based on embedding a computational domain into a background mesh that is not required to match the boundary leading to so-called cut elements at the boundary. Adding stabilization terms, we can control the variation of the discrete functions close to the boundary, which allows us to prove stability, condition number estimates, and optimal order a priori error estimates. Alternatively, we may use a discrete extension operator and solve the problem in a subspace of the finite element space where the unstable degrees of freedom are eliminated in such a way that optimal order approximation bounds are retained. These two approaches, the first weak and the second strong, have the same goal: to stabilize the method but appear very different at first glance. We show that the definition of stabilization terms, added to the weak statement, may be generalized in two ways: (1) The stabilized quantity may be some functional of the discrete function, for instance, finite element degrees freedom. This allows us to stabilize the unstable modes more precisely than standard approaches, which may be viewed as element-based. (2) The choice of elements that are connected. Typically, face neighbors, or connected patches are used, but we may stabilize by connecting elements intersecting the boundary to an element within a distance proportional to the mesh parameter. We show that the generalized stabilization form fits into the standard abstract requirements, and as a consequence, we obtain stable and optimal order convergent methods for second-order elliptic problems. We also show that for a robust design of the ghost penalty, one may let the stabilization parameter tend to infinity without introducing locking. The limit corresponds to strong enforcement of certain algebraic constraints, which are identical to constraints implemented in specific extension operator frameworks. This illustrates the very close connection between stabilization and extension approaches.