Andrew T. Barker

Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling

We introduce and study numerically a scalable parallel finite element solver for the simulation of blood flow in compliant arteries. The incompressible Navier-Stokes equations are used to model the fluid and coupled to an incompressible linear elastic model for the blood vessel walls. Our method features an unstructured dynamic mesh capable of modeling complicated geometries, an arbitrary Lagrangian-Eulerian framework that allows for large displacements of the moving fluid domain, monolithic coupling between the fluid and structure equations, and fully implicit time discretization. Simulations based on blood vessel geometries derived from patient-specific clinical data are performed on large supercomputers using scalable Newton-Krylov algorithms preconditioned with an overlapping restricted additive Schwarz method that preconditions the entire fluid-structure system together. The algorithm is shown to be robust and scalable for a variety of physical parameters, scaling to hundreds of processors and millions of unknowns.

Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.

Two-Level Newton and Hybrid Schwarz Preconditioners for Fluid-Structure Interaction

We introduce and study numerically a two-level Schwarz preconditioner for Newton-Krylov methods for fluid-structure interaction, with special consideration of the application area of simulating blood flow. Our approach monolithically couples the fluid to the structure on both fine and coarse grids and in the subdomain solves, insuring that there is multiphysics coupling during all aspects of the algorithm. The fluid-structure system is discretized on unstructured nonnested meshes, with an overlapping additive domain decomposition on both coarse and fine levels and multiplicative Schwarz preconditioning between levels. We investigate the effect of different coarse discretization sizes, solver stopping criteria, and overlap size, and we demonstrate that the method is robust to physical parameters including the structures Youngs modulus and the timestep size. Finally, we show effective preconditioning of the complicated coupled system, with nearly perfect weak scaling to a thousand processors and millions of unknowns.

A One-Level Additive Schwarz Preconditioner for a Discontinuous Petrov–Galerkin Method

Discontinuous Petrov–Galerkin (DPG) methods are new discontinuous Galerkin methods [3–8] with interesting properties. In this article we consider a domain decomposition preconditioner for a DPG method for the Poisson problem.

Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems

Abstract We propose and analyze overlapping two-level additive Schwarz preconditioners for the local discontinuous Galerkin discretization. We prove that the condition number of the preconditioned system is bounded by C[1 + (H/δ)], where H represents the coarse mesh size, δ measures the overlap among the subdomains, and the constant C is independent of H, δ, the fine mesh size h and the number of subdomains Ns . Numerical results are presented showing the scalability of the method.

A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

We present a mixed finite element method for the steady-state Stokes equations where the discrete bilinear form for the velocity is obtained by a weakly over-penalized symmetric interior penalty approach. We show that this mixed finite element method is inf-sup stable and has optimal convergence rates in both the energy norm and the

A Combined Preconditioning Strategy for Nonsymmetric Systems

L_2

Evolutionary Stability in the Traveler's Dilemma

L2 norm on meshes that can contain hanging nodes. We present numerical experiments illustrating these results, explore a very simple adaptive algorithm that uses meshes with hanging nodes, and introduce a simple but scalable parallel solver for the method.