Adaptive finite element methods for parabolic problems. I.: a linear model problem
SIAM Journal on Numerical Analysis
High-order accurate discontinuous finite element solution of the 2D Euler equations
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations
Journal of Computational Physics
On some aspects of the discontinuous Galerkin finite element method for conservation laws
Mathematics and Computers in Simulation - MODELLING 2001 - Second IMACS conference on mathematical modelling and computational methods in mechanics, physics, biomechanics and geodynamics
Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows
Journal of Computational Physics
Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Journal of Computational Physics
"Natural norm" a posteriori error estimators for reduced basis approximations
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations
Journal of Computational Physics
Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems
SIAM Journal on Control and Optimization
Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations
SIAM Journal on Scientific Computing
Error estimation and adaptation for functional outputs in time-dependent flow problems
Journal of Computational Physics
Adaptive time-step with anisotropic meshing for incompressible flows
Journal of Computational Physics
Output-based mesh adaptation for high order Navier-Stokes simulations on deformable domains
Journal of Computational Physics
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This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space-time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space-time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier-Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space-time meshes are shown to be relatively insensitive to the starting mesh.