The present article is an overview of some mathematical results, whichprovide elements of rigorous basis for some multiscalecomputations in materials science. The emphasis is laid upon atomisticto continuum limits for crystalline materials. Various mathematicalapproaches are addressed. Thesetting is stationary. The relation to existing techniques used in the engineeringliterature is investigated.

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arenathat follow this design methodology providing extensible, reusable,and flexible software while staying competitive to traditionallydesigned point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call “concept oriented design”. We apply this design methodology to Petrov-Galerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods (FEM). We show as a particular example the implementation of hp-FEM using the library with specialattention to the handling of inconsistent meshes.

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperatureθ and an ODE with double obstaclesfor phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requiresthe stability constraint τ ≤ ε. Conforming piecewiselinear finite elements over highly graded simplicial mesheswith parameter h are further employed for space discretization.A posteriori errorestimates are derived for both unknowns θ and χ, whichexhibit the correct asymptotic order in terms of ε, h andτ. This result circumvents the use of duality, which does noteven apply in this context.Several numerical experiments illustrate the reliability of theestimators and document the excellent performance of the ensuingadaptive method.