Through the past three decades gas bearings have found way into an increasing number of industrial applications within high speed rotating machinery. Especially the compliant foil type of bearings has gained large popularity. Much theoretical and experimental work has been conducted on the compliant foil bearings, and the understanding of their dynamic behaviour is growing. However, practical design involving these bearings are still associated with a large degree of trial and error. This study aims at establishing an accurate mathematical model, to calculate the pressure, film height and dynamic coefficients, of the compliant foil bearing together with an efficient solution method, which can be easily adopted and implemented by mechanical engineers. A theoretical model of a radial compliant foil bearing that incorporates compressibility of the lubricating gas and flexibility/compliance of the foil structure is presented. The compliance of the foil structure is incorporated implicitly in the Reynolds equation which is accomplished through a modification of the film gap function . The resulting non-linear equation is perturbed and solved by use of the finite element method following a Bubnow-Galerkin approach. This constitutes the main original contribution of this work, considering the fact that the finite difference method is commonly used and thouroughly investigated in the literature. The finite element method leads to a set of non-linear equations for the static fluid film pressure (zeroth order) which can be solved by an iterative approach, where the pressure field is the converging parameter. The equations for the dynamic pressures (first order) becomes linear and can be solved directly to obtain the linearised stiffness and damping coefficients of the bearing. The influence of explicit and implicit boundary conditions are also investigated. Theoretical results for pressures, shaft equilibrium positions and film thickness are presented and compared to experimental results [17, 19]. A good agreement between experimental and theoretical results are found for large loads. For lower loads, some discrepancies are observed and discussed in details. The dynamic stiffness and damping coefficients are calculated and compared to theoretical results reported in . A good agreement are observed for both stiffness and damping coefficients.