×

zbMATH — the first resource for mathematics

On the stability of DPG formulations of transport equations. (English) Zbl 1383.65102
Summary: In this paper we formulate and analyze a discontinuous Petrov-Galerkin (DPG) 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.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L02 First-order hyperbolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
iFEM
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Barrett, J. W.; Morton, K. W., Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Engrg., 45, 1-3, 97-122, (1984) · Zbl 0562.76086
[2] Broersen, Dirk; Stevenson, Rob, A robust Petrov-Galerkin discretisation of convection-diffusion equations, Comput. Math. Appl., 68, 11, 1605-1618, (2014) · Zbl 1364.65191
[3] Broersen, Dirk; Stevenson, Rob P., A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form, IMA J. Numer. Anal., 35, 1, 39-73, (2015) · Zbl 1311.65144
[4] Carstensen, C.; Demkowicz, L.; Gopalakrishnan, J., Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl., 72, 3, 494-522, (2016) · Zbl 1359.65249
[5] [C09]ifem L. Chen, iFEM: An integrated finite element method package in MATLAB, Technical Report, University of California at Irvine, 2009.
[6] Cohen, Albert; Dahmen, Wolfgang; Welper, Gerrit, Adaptivity and variational stabilization for convection-diffusion equations, ESAIM Math. Model. Numer. Anal., 46, 5, 1247-1273, (2012) · Zbl 1270.65065
[7] Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations, 27, 1, 70-105, (2011) · Zbl 1208.65164
[8] Dahmen, Wolfgang; Huang, Chunyan; Schwab, Christoph; Welper, Gerrit, Adaptive Petrov-Galerkin methods for first order transport equations, SIAM J. Numer. Anal., 50, 5, 2420-2445, (2012) · Zbl 1260.65091
[9] Dahmen, Wolfgang; Plesken, Christian; Welper, Gerrit, Double greedy algorithms: reduced basis methods for transport dominated problems, ESAIM Math. Model. Numer. Anal., 48, 3, 623-663, (2014) · Zbl 1291.65339
[10] De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F.; Olson, Luke, Least-squares finite element methods and algebraic multigrid solvers for linear hyperbolic PDEs, SIAM J. Sci. Comput., 26, 1, 31-54, (2004) · Zbl 1105.65346
[11] Gopalakrishnan, Jay; Monk, Peter; Sep\'ulveda, Paulina, A tent pitching scheme motivated by Friedrichs theory, Comput. Math. Appl., 70, 5, 1114-1135, (2015)
[12] Gopalakrishnan, J.; Qiu, W., An analysis of the practical DPG method, Math. Comp., 83, 286, 537-552, (2014) · Zbl 1282.65154
[13] Heuer, Norbert; Karkulik, Michael; Sayas, Francisco-Javier, Note on discontinuous trace approximation in the practical DPG method, Comput. Math. Appl., 68, 11, 1562-1568, (2014) · Zbl 1364.65250
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.