Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection
Journal of Computational Physics
Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws
Journal of Computational Physics
Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations
SIAM Journal on Scientific Computing
Goal-oriented a posteriori error estimates for transport problems
Mathematics and Computers in Simulation
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The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems. The Galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints. The localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions. A nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible. The developed simulation tools are applied to a linear convection problem in two space dimensions.