Many problems involving fluid flow can now be simulated numerically, providing a useful predictive tool for a wide range of engineering applications. Of particular interest in this thesis are computational methods for solving the problem of compressible fluid flow around aerodynamic configurations.\ud \ud A finite element method is presented for solving the compressible Navier-Stokes equations in two dimensions on unstructured meshes. The method is stablized by the addition of a least-squares operator (an inexpensive simplification of the Galerkin least-squares method), leading to solutions free of spurious oscillations. Convergence to steady state is reached via a backward Euler time-stepping scheme, and the use of local time-steps allows convergence to be accelerated. The choice of both the nonlinear solver, which is employed to solve the algebraic system arising at each time-step, and the iterative method used within this solver, is fully discussed, along with an inexpensive technique for approximating the Jacobian matrix.\ud \ud In order to obtain accurate solutions more effectively, the idea of adapting the mesh is investigated, and two distinct methods of mesh refinement are described in detail. These are the addition of nodes to the mesh in regions determined by an error indicator (h - refinement ) and the local repositioning of existing nodes using the value of this error indicator across neighbouring elements (r - refinement). As well as considering these adaptive techniques separately, we introduce an original algorithm which combines the two ideas, with results indicating that this combination is an effective approach. The example problems used consist mainly of steady transonic flow at low to moderate Reynolds numbers.\ud \ud Transient flow problems are also considered, and we examine the difficulties which occur when the method of lines is used as a solution technique and h-refinement (including derefinement of elements) is carried out
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