Parallel Eulerian Methods for Flows with Elastic Membranes∗

We propose and study parallel numerical algorithms for simulation of flows with elastic membranes, such as red blood cells. The challenges of modeling such flows include accurately resolving the interface motion and enforcing the appropriate interface mechanics. We propose a phase field Eulerian method for implicitly capturing the location of the dynamic interface between two fluids. In addition, for flows with elastic membranes separating the fluids, we present a phase field formulation for the membrane stresses. The method we propose avoids the computational geometry complexities associated with fully Lagrangian or front-tracking Eulerian methods. These complexities are particularly pronounced on parallel computers. The phase field formulation is discretized by a combined continuous/discontinuous Galerkin method, which lends itself to an inherently parallel implementation. We present a block Schur-complement preconditioner for the coupled membrane-flow system that neutralizes the ill-conditioning due to disparate material properties. Parallel performance results demonstrate the efficacy of this preconditioner.