Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetrizable hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree and . We apply these asymptotic results to show that projections of the error are pointwise -superconvergent. We solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.

[1]  Joseph E. Flaherty,et al.  Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems , 2003, Adv. Comput. Math..

[2]  Bernardo Cockburn.  A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws , 1999 .

[3]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[4]  T. C. Massey.  SUPERCONVERGENCE OF DISCONTINUOUS FINITE ELEMENT SOLUTIONS FOR NONLINEAR HYPERBOLIC PROBLEMS , 2022 .

[5]  Chi-Wang Shu,et al.  Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension , 2010, SIAM J. Numer. Anal..

[6]  Richard H. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer , 1984 .

[7]  Slimane Adjerid,et al.  Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem , 2006 .

[8]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[9]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[10]  Ralf Hartmann,et al.  Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws , 2002, SIAM J. Sci. Comput..

[11]  T. Weinhart,et al.  Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems , 2009 .

[12]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[13]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[14]  Bernardo Cockburn,et al.  Error estimates for finite element methods for scalar conservation laws , 1996 .

[15]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[16]  Mahboub Baccouch,et al.  The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis , 2007, J. Sci. Comput..

[17]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[18]  Chi-Wang Shu,et al.  Superconvergence and time evolution of discontinuous Galerkin finite element solutions , 2008, J. Comput. Phys..

[19]  Sylvie Benzoni-Gavage,et al.  Multi-dimensional hyperbolic partial differential equations , 2006 .

[20]  Mahboub Baccouch,et al.  The Discontinuous Galerkin Method for Two-dimensional Hyperbolic Problems Part II: A Posteriori Error Estimation , 2009, J. Sci. Comput..

[21]  Slimane Adjerid,et al.  A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems , 2002 .

[22]  Timothy J. Barth,et al.  A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems , 2000 .

[23]  Yingda Cheng,et al.  Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations , 2009 .

[24]  Uriel G. Rothblum,et al.  A Representation of the Drazin Inverse and Characterizations of the Index , 1976 .

[25]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[26]  Mahboub Baccouch,et al.  Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem , 2010 .