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On the stability of DPG formulations of transport equations

Authors: D. Broersen, W. Dahmen and R. P. Stevenson
Journal: Math. Comp. 87 (2018), 1051-1082
MSC (2010): Primary 65N12, 65N30, 35A15, 35F05
Published electronically: September 7, 2017
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Abstract: In this paper we formulate and analyze a Discontinuous Petrov-Galerkin formulation of linear transport equations with variable convection fields. We show that a corresponding infinite dimensional mesh-dependent variational formulation, in which besides the principal field its trace on the mesh skeleton is also an unknown, is uniformly stable with respect to the mesh, where the test space is a certain product space over the underlying domain partition.

Our main result then states the following. For piecewise polynomial trial spaces of degree $ m$, we show under mild assumptions on the convection field that piecewise polynomial test spaces of degree $ m+1$ over a refinement of the primal partition with uniformly bounded refinement depth give rise to uniformly (with respect to the mesh size) stable Petrov-Galerkin discretizations. The partitions are required to be shape regular but need not be quasi-uniform. An important startup ingredient is that for a constant convection field one can identify the exact optimal test functions with respect to a suitably modified but uniformly equivalent broken test space norm as piecewise polynomials. These test functions are then varied towards simpler and stably computable near-optimal test functions for which the above result is derived via a perturbation analysis. We conclude indicating some consequences of the results that will be treated in forthcoming work.

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Additional Information

D. Broersen
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

W. Dahmen
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

R. P. Stevenson
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Keywords: Discontinuous Petrov-Galerkin formulation of transport equations, optimal and near-optimal test spaces, stability
Received by editor(s): October 7, 2015
Received by editor(s) in revised form: September 6, 2016, and November 2, 2016
Published electronically: September 7, 2017
Additional Notes: The first author was supported by the Netherlands Organization for Scientific Research (NWO) under contract no. 613.001.109
The second author was supported in part by the DFG SFB-Transregio 40, by the DFG Research Group 1779, and the Excellence Initiative of the German Federal and State Governments
Article copyright: © Copyright 2017 American Mathematical Society

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