Both approaches are used to deal with isotropic as well as anisotropic computational grids. More recently, the interest has moved towards anisotropic mesh adaptation as more efficient in tackling problems characterized by directional features, e.g. shocks, boundary and internal layers, etc.
In this communication we focus on theoretically sound anisotropic adaptation techniques. The abstract framework has been the derivation of anisotropic interpolation error estimates for piecewise linear finite elements. This has been obtained by exploiting the spectral properties of the standard affine mapping between the reference and the general mesh element. Then we have merged these estimates with the dual-based a posteriori error analysis proposed by R. Rannacher and R. Becker. This approach turns out to be suitable for a goal-oriented adaptivity as it allows us to control energy norms as well as suitable functionals of the discretisation error. For instance, we can deal with meaningful physical quantities such as mean and local values, point-wise stresses or concentrations, the lift and drag around a body in a flow, etc. On the other hand, the main drawback of this analysis is the more demanding computational cost, usually higher than the one required by a standard residual-based approach.
In this talk we provide as examples of this theory advection-diffusion, the Stokes and Oseen problems and we validate our results on some numerical test cases.