Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations
Journal of Computational Physics
Implicit-explicit time stepping with spatial discontinuous finite elements
Applied Numerical Mathematics
Journal of Computational Physics
Adaptive unstructured volume remeshing - I: The method
Journal of Computational Physics
Even-odd goal-oriented a posteriori error estimation for elliptic problems
Applied Numerical Mathematics
Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis
International Journal of Computing Science and Mathematics
A new a posteriori error estimate for convection-reaction-diffusion problems
Journal of Computational and Applied Mathematics
Journal of Scientific Computing
Applied Numerical Mathematics
Even--odd goal-oriented a posteriori error estimation for elliptic problems
Applied Numerical Mathematics
Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws
Applied Numerical Mathematics
Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems
Journal of Computational and Applied Mathematics
Goal-oriented a posteriori error estimates for transport problems
Mathematics and Computers in Simulation
Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem
Applied Numerical Mathematics
Journal of Computational Physics
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Adaptive Timestep Control for Nonstationary Solutions of the Euler Equations
SIAM Journal on Scientific Computing
New adaptive artificial viscosity method for hyperbolic systems of conservation laws
Journal of Computational Physics
Parallel mesh adaptive techniques illustrated with complex compressible flow simulations
Modelling and Simulation in Engineering
Advances in Computational Mathematics
Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems
Applied Numerical Mathematics
A posteriori error estimation for the Lax-Wendroff finite difference scheme
Journal of Computational and Applied Mathematics
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We consider the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws. In particular, we discuss the question of error estimation for general target functionals of the solution; typical examples include the outflow flux, local average and pointwise value, as well as the lift and drag coefficients of a body immersed in an inviscid fluid. By employing a duality argument, we derive so-called weighted or Type I a posteriori error bounds; these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated. Based on the resulting approximate Type I bound, we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest. The performance of the proposed adaptive strategy and the quality of the approximate Type I a posteriori error bound is illustrated by a series of numerical experiments. In particular, we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ Type II error indicators.