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.