Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws
R. Hartmann and P. Houston
Abstract:
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
linear and nonlinear 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; in these error estimates
the element--residuals are multiplied by local weights involving
the solution of a certain dual problem. Based on these a posteriori
bounds, we design and implement the corresponding adaptive algorithm
to ensure efficient and reliable control of the error in the
computed functional. The theoretical results are illustrated by a series of
numerical experiments. In particular, we demonstrate the superiority
of the proposed approach over standard mesh refinement algorithms which employ
ad hoc error indicators.