Joint work with C.E. Goodyer R. Fairlie at Leeds, Chris Johnson of the SCI Institute at the University of Utah, and L.E. Scales of Shell Research and Technology Centre, Thornton, Chester, UK The solution of real life computational engineering problems poses many challenges for adaptive Mesh Refinement (AMR). The AMR requirements of two such problems will be considered in the light of recent developments in both theory and algorithms. The two motivating problems are Elasto-Hydrodynamic Lubrication (EHL) as being modeled at Leeds and the explosive container problem currently being solved by the CSAFE centre at the University of Utah. In the case of the EHL example, although the modelling of lubrication problems extends back to the hydrodynamic equation of Reynolds, for very highly loaded cases (from 0.25 GPa) the contacts themselves deform, defining elastohydrodynamic lubrication (EHL). These cases are important to industry for the design of new oils and new components and pose a formidable computational engineering challenge. Coupling the dense matrices of the elastic deformation of the surfaces with the highly non-linear equations of the pressure distribution and lubricant rheology leads to an intensive computation requiring at least 100 million grid points to resolve micro-EHL problems (which are generally also transient) with realistic surface roughness. Although the CSAFE problem is perhaps an order of magnitude more challenging still, both problems have a number of common characteristics:

(i) complex multi-scale phenomena and experimental physical models

(ii) an inherently transient nature with multiple space and time scales

(iii) the need for parallel computing and large numbers of grid points

(iv) accuracy requirements specified in terms of particular quantities of interest.

(v) the need to use adaptive multigrid methods on cartesian type meshes.

(vi) the end-user desire for an integrated problem solving environment

These characteristics will be described in relation to the problems under consideration and used to motivate an examination of existing techniques for transient error estimation and for determining the error in quantities of interest. The theoretical and algorithmic issues related to current AMR techniques will be considered and the techniques applied to current lubrication problems. The challenge of applying such techniques to the CSAFE problem will be considered.