Open Access

We present an immersed boundary method for the solution of elliptic interface problems with discontinuous coefficients which provides a second-order approximation of the solution. The proposed method can be categorised as an extended or enriched finite element method. In contrast to other extended FEM approaches, the new shape functions get projected in order to satisfy the Kronecker-delta property with respect to the interface. The resulting combination of projection and restriction was already derived in Höllbacher and Wittum (TBA, 2019a) for application to particulate flows. The crucial benefits are the preservation of the symmetry and positive definiteness of the continuous bilinear operator. Besides, no additional stabilisation terms are necessary. Furthermore, since our enrichment can be interpreted as adaptive mesh refinement, the standard integration schemes can be applied on the cut elements. Finally, small cut elements do not impair the condition of the scheme and we propose a simple procedure to ensure good conditioning independent of the location of the interface. The stability and convergence of the solution will be proven and the numerical tests demonstrate optimal order of convergence.

Highlights • Monolithic scheme for particulate flows preventing an oscillating pressure along the interface. • The choice of enriching shape functions is driven by the properties of its gradient instead of its value. • The choice of enriching shape functions inherits a natural stabilization on small cut elements. Abstract We present gradient-consistent enriched finite element spaces for the simulation of free particles in a fluid. This involves forces being exchanged between the particles and the fluid at the interface. In an earlier work [23] we derived a monolithic scheme which includes the interaction forces into the Navier-Stokes equations by means of a fictitious domain like strategy. Due to an inexact approximation of the interface oscillations of the pressure along the interface were observed. In multiphase flows oscillations and spurious velocities are a common issue. The surface force term yields a jump in the pressure and therefore the oscillations are usually resolved by extending the spaces on cut elements in order to resolve the discontinuity. For the construction of the enriched spaces proposed in this paper we exploit the Petrov-Galerkin formulation of the vertex-centered finite volume method (PG-FVM), as already investigated in [23]. From the perspective of the finite volume scheme we argue that wrong discrete normal directions at the interface are the origin of the oscillations. The new perspective of normal vectors suggests to look at gradients rather than values of the enriching shape functions. The crucial parameter of the enrichment functions therefore is the gradient of the shape functions and especially the one of the test space. The distinguishing feature of our construction therefore is an enrichment that is based on the choice of shape functions with consistent gradients. These derivations finally yield a fitted scheme for the immersed interface. We further propose a strategy ensuring a well-conditioned system independent of the location of the interface. The enriched spaces can be used within any existing finite element discretization for the Navier-Stokes equation. Our numerical tests were conducted using the PG-FVM. We demonstrate that the enriched spaces are able to eliminate the oscillations.