DPG approximation of eigenvalue problems

In this paper, the discontinuous Petrov--Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultra weak formulations of the problem and prove the convergence together with a priori error estimates. Moreover, we propose two possible error estimators and perform the corresponding a posteriori error analysis. The theoretical results are confirmed numerically and it is shown that the error estimators can be used to design an optimally convergent adaptive scheme.

[1]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[2]  Carsten Carstensen,et al.  Breaking spaces and forms for the DPG method and applications including Maxwell equations , 2015, Comput. Math. Appl..

[3]  Ignacio Muga,et al.  Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for Helmholtz Equation , 2013, SIAM J. Sci. Comput..

[4]  Daniele Boffi,et al.  First order least-squares formulations for eigenvalue problems , 2020, IMA Journal of Numerical Analysis.

[5]  Leszek F. Demkowicz,et al.  A primal DPG method without a first-order reformulation , 2013, Comput. Math. Appl..

[6]  F. Hellwig Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods , 2019 .

[7]  Thomas Führer Superconvergence in a DPG method for an ultra-weak formulation , 2018, Comput. Math. Appl..

[8]  Jay Gopalakrishnan,et al.  Convergence rates of the DPG method with reduced test space degree , 2014, Comput. Math. Appl..

[9]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[10]  Jay Gopalakrishnan A CLASS OF DISCONTINUOUS PETROV-GALERKIN METHODS. PART II: OPTIMAL TEST FUNCTIONS , 2009 .

[11]  Victor M. Calo,et al.  A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D , 2011, J. Comput. Phys..

[12]  Nathan V. Roberts,et al.  The DPG method for the Stokes problem , 2014, Comput. Math. Appl..

[13]  Weifeng Qiu,et al.  A locking-free hp DPG method for linear elasticity with symmetric stresses , 2012, Numerische Mathematik.

[14]  Weifeng Qiu,et al.  An analysis of the practical DPG method , 2011, Math. Comput..

[15]  Carsten Carstensen,et al.  A Posteriori Error Control for DPG Methods , 2014, SIAM J. Numer. Anal..

[16]  Robert D. Moser,et al.  A DPG method for steady viscous compressible flow , 2014 .

[17]  Victor M. Calo,et al.  Analysis of the discontinuous Petrov-Galerkin method with optimal test functions for the Reissner-Mindlin plate bending model , 2013, Comput. Math. Appl..

[18]  Daniele Boffi,et al.  Least-squares for linear elasticity eigenvalue problem , 2020, ArXiv.

[19]  Omar Ghattas,et al.  A Unified Discontinuous Petrov-Galerkin Method and Its Analysis for Friedrichs' Systems , 2013, SIAM J. Numer. Anal..

[20]  Carsten Carstensen,et al.  Low-order dPG-FEM for an elliptic PDE , 2014, Comput. Math. Appl..

[21]  Leszek Demkowicz,et al.  A Class of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation , 2010 .

[22]  Joachim Sch NETGEN An advancing front 2D/3D-mesh generator based on abstract rules , 1997 .

[23]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

[24]  Nathan V. Roberts,et al.  A discontinuous Petrov-Galerkin methodology for adaptive solutions to the incompressible Navier-Stokes equations , 2015, J. Comput. Phys..