Efficient High-Frequency Time-Harmonic Wave Propagation Solvers Using Discontinuous Petrov-Galerkin Methods

Project Overview

The purpose of this project is to investigate new ways of solving high-frequency wave propagation problems based on the Discontinuous Petrov-Galerkin (DPG) method. With traditional discretization methods, these simulations are extremely challenging because of two major issues: instability and indefiniteness of the linear systems. Consequently, traditional solvers simply break down. On the other hand, the DPG method, overcomes both issues. It is a non-standard minimum residual method that promises high accuracy, unconditional stability, and definite linear systems. This project investigates the feasibility of solving large-scale wave-propagation problems by integrating modern elliptic solvers with DPG discretizations. For that we developed DPG formulations in MFEM and studied the efficiency of algebraic solvers from Hypre for the solution of high-frequency wave-propagation problems. We focus on acoustics and electromagnetic problems and provide a comparison with existing state-of-the-art wave-propagation solvers.

This feasibility study turned out to be very successful. We were able to answer the feasibility question: "Can we solve large-scale wave propagation problems by integrating state-of-the-art elliptic solvers with DPG discretizations?". From our results we conclude that the answer is positive. It is evident that combining state-of -the-art elliptic solvers with DPG discretizations creates a new and improved paradigm for solving high-frequency wave-propagation problems. We emphasize that the newly developed solver has a black-box nature, i.e., it is fully algebraic and this makes it unique compared to other sophisticated wave solvers. It delivers faster solution times for problems that are ideal for existing wave solvers, but at the same time it can be applied to problems with more complicated settings, such as unstructured and AMR meshes, with favorable convergence behavior. In our opinion, these results can be further improved by employing GPU architectures and matrix-free solvers, research that we intend to suggest in future project proposals.

Mission Impact

Our project directly supports the LLNL core competency high-performance computing. This work will be beneficial primarily to applications areas that involve high-frequency time-harmonic wave-propagation simulations, such as plasma fusion and non-destructive testing, which are of interest to SciDAC, LLNL FES and LLNL NCI, respectively. However, the foundational DPG infrastructure we developed can also benefit simulations in other application areas, such as convection-diffusion transport, fluid flow, and engineering design. Scientists in LLNL's Engineering Directorate have already expressed interest in applying DPG to hypersonic flow and turbulence simulations. In the DOE there are also applications of DPG to electromagnetic simulations at Sandia National Laboratories.