Adjoint error estimation and adaptivity for hyperbolic problems - Houston, Paul (Elsevier / North Holland, 2017)
In this article we present an overview of a posteriori error estimation and adaptive mesh design for hyperbolic/nearly-hyperbolic problems. 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 weighted Type I and unweighted Type II bounds may be established. Here, the relative advantages of these two approaches are discussed in detail, together with the construction of appropriate dual problems that ensure optimality of the resulting bounds. The exploitation of general adaptive refinement strategies based on employing isotropic and anisotropic h- and hp-refinement will be discussed. Applications of this general theory to eigenvalue problems and bifurcation problems will also be presented.