In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent. The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.
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