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Advanced Numerical Methods in the Mathematical Sciences

Truman Ellis, University of Texas at Austin
Space-Time Discontinuous Petrov-Galerkin Finite Elements for Fluid Flow

Abstract

The discontinuous Petrov-Galerkin method is a novel finite element framework with exceptional stability and adaptivity properties. The DPG framework can be used to derive stable discretizations of any well-posed variational formulation and has been successfully applied to problems such as heat conduction, time-harmonic Helmholtz, Maxwell's equations, linear elasticity and plate problems, Stokes flow, and both incompressible and compressible Navier-Stokes. In contrast to many other numerical methods, DPG does not suffer from a pre-asymptotic regime (unstable behavior on coarse meshes). This means that a simulation can be initiated at the coarsest scale possible while automatic adaptivity resolves solution features based on a robust, built-in measure of residual error. DPG is intensely locally compute intensive with significant embarrassingly parallel computations done both pre- and post-global solve. We present preliminary work on a space-time DPG formulation that enables automatic local time stepping and a kind of parallel-in-time integration.