Robust DPG Methods for Transient Convection-Diffusion
We introduce two robust DPG methods for transient convection-diffusion problems. Once a variational formulation is selected, the choice of test norm critically influences the quality of a particular DPG method. It is desirable that a test norm produce con
- PDF / 744,693 Bytes
- 25 Pages / 439.36 x 666.15 pts Page_size
- 4 Downloads / 153 Views
Abstract We introduce two robust DPG methods for transient convection-diffusion problems. Once a variational formulation is selected, the choice of test norm critically influences the quality of a particular DPG method. It is desirable that a test norm produce convergence of the solution in a norm equivalent to L2 while producing optimal test functions that can be accurately computed and maintaining good conditioning of the optimal test function solve on highly adaptive meshes. Two such robust norms are introduced and proven to guarantee close to L2 convergence of the primary solution variable. Numerical experiments demonstrate robust convergence of the two methods.
1 Introduction The discontinuous Petrov-Galerkin finite element method presents an attractive new framework for developing robust numerical methods for computational mechanics. DPG contains the promise of being an automated scientific computing technology— it provides stability for any variational formulation, optimal convergence rates in a user-defined norm, virtually no pre-asymptotic stability issues on coarse meshes, and a measure of the error residual which can be used to robustly drive adaptivity. The method also delivers Hermitian positive definite stiffness matrices for any problem, weak enforcement of boundary conditions, and several other attractive properties [9, 11, 12]. For the most recent review of DPG, see [13]. The process of developing robust DPG methods for steady convection-diffusion was explored
T. Ellis • L. Demkowicz () Institute for Computational Engineering and Sciences, University of Texas at Austin, 201 East 24th St, Stop C0200, Austin, TX 78712, USA e-mail: [email protected]; [email protected] J. Chan Department of Mathematics, Virginia Tech, 460 McBryde Hall, Virginia Tech 225 Stanger Street, Blacksburg, VA 24061-0123, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 G.R. Barrenechea et al. (eds.), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering 114, DOI 10.1007/978-3-319-41640-3_6
179
180
T. Ellis et al.
in [7, 14]. In the sense, the main challenge is to come up with a correct test norm. The residual is measured in the dual test norm, and the DPG method minimizes the residual. The residual can be interpreted as a special energy norm. In other words, the DPG method delivers an orthogonal projection in the energy norm. The task is especially challenging for singular perturbation problems. Given a trial norm, we strive to determine a quasi-optimal test norm such that the corresponding energy norm is robustly equivalent to the trial norm of choice. An additional difficulty comes from the fact that the optimal test functions should be easily approximated with a simple enrichment strategy. For convection dominated diffusion, this means that the test functions should not develop boundary layers. The task of determining the quasi optimal test norm (we call it a rob
Data Loading...
