Past Event: Oden Institute Seminar

An Update on Discontinuous Petrov-Galerkin (Dpg) FE Method with Optimal Test Functions

Leszek Demkowicz, Professor, ICES

Abstract

I will provide a progress report on the DPG method co-invented with Jay Gopalakrishnan over a year ago. The method guarantees automatically the discrete stability provided the continuous problem is well posed, and can be applied to any linear system of PDEs and any choice of trial functions. In particular, the methodology extends dramatically the applicability of hp approximations. The DPG method builds on two fundamental ideas: - a Petrov-Galerkin method with optimal test functions, - a discontinuous Petrov-Galerkin formulation based on the so-called ultra-weak variational hybrid formulation. We will use linear acoustics and convection-dominated diffusion as model problems to present the main concepts and illustrate it with 1D and 2D numerical results. Time permitting, I will overview a number of other applications for which we have collected some numerical experience including: 1D Burgers and compressible Navier-Stokes equations (shocks) Timoshenko beam and axisymmetric shells (locking, boundary layers) 2D linear elasticity (mixed formulation, singularities) 2D convection and systems of hyperbolic equations. We have recently managed to obtain our very first proof of well-posedness (inf-sup condition) for multi-dimensional Laplace and convection-dominated diffusion [8]. I will state the result but postpone a detailed presentation of the proof to another presentation. The presented methodology incorporates the following features: The problem of interest is formulated as a system of first order PDE's in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the integration by parts is done over individual elements. As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns (a hybrid method). For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element. The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory. The resulting stiffness matrix is always hermitian and positive- definite. In fact, the method can be interpreted as least-squares applied to a preconditioned version of the problem. For singular perturbation problems, by selecting right norms for test functions, we can obtain stability properties uniform not only with respect to discretization parameters but also with respect to the perturbation parameter (diffusion constant, Reynolds number, beam or shell thickness, wave number). In other words, the resulting discretization is robust.