Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h2p+1) compared to a global rate of O(hp+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.

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

[2]  Jean-François Remacle,et al.  An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems , 2003, SIAM Rev..

[3]  J. Tinsley Oden,et al.  A Posteriori Error Estimation , 2002 .

[4]  J. Flaherty,et al.  Parallel, adaptive finite element methods for conservation laws , 1994 .

[5]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

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

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

[8]  PAUL HOUSTON,et al.  Stabilized hp-Finite Element Methods for First-Order Hyperbolic Problems , 2000, SIAM J. Numer. Anal..

[9]  Joseph E. Flaherty,et al.  Software for the parallel adaptive solution of conservation laws by discontinous Galerkin methods. , 2000 .

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[12]  Endre Süli,et al.  A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems , 1997 .

[13]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[14]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[15]  Endre Süli,et al.  Hp-DGFEM for Partial Di erential Equations with Nonegative Characteristic Form , 1999 .

[16]  Boleslaw K. Szymanski,et al.  Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws , 1997, J. Parallel Distributed Comput..

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

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

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

[20]  Jean-François Remacle,et al.  Aspects of discontinuous Galerkin methods for hyperbolic conservation laws , 2002 .

[21]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[22]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..