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The adaptive implicit-explicit (AIE) approach is presented for the finite-element solution of various problems in computational fluid mechanics. In the AIE approach, the elements are dynamically (adaptively) arranged into differently treated groups. The differences in treatment could be based on considerations such as the cost efficiency, the type of spatial or temporal discretization employed, the choice of field equations, etc. Several numerical tests are performed to demonstrate that this approach can achieve substantial savings in CPU time and memory.

Element-by-element approximate factorization, implicit-explicit and adaptive implicit-explicit approximation procedures are presented for the finite-element formulations of large-scale fluid dynamics problems. The element-by-element approximation scheme totally eliminates the need for formation, storage and inversion of large global matrices. Implicit-explicit schemes, which are approximations to implicit schemes, substantially reduce the computational burden associated with large global matrices. In the adaptive implicit-explicit scheme, the implicit elements are selected dynamically based on element level stability and accuracy considerations. This scheme provides implicit refinement where it is needed. The methods are applied to various problems governed by the convection-diffusion and incompressible Navier-Stokes equations. In all cases studied, the results obtained are indistinguishable from those obtained by the implicit formulations.

Multi-dimensional generalizations of a one-dimensional finite element shock capturing scheme are proposed. A scalar model problem is used to emphasize that 'preferred directions' are important in multi-dimensional applications. Schemes are developed for the two-dimensional Euler equations. One, based upon characteristics, employs the Mach lines and streamlines as preferred directions.

Element-by-element approximate factorization procedures are proposed for solving the large finite element equation systems which arise in computational mechanics. A variety of techniques are compared on problems of structural mechanics, heat conduction and fluid mechanics. The results obtained suggest considerable potential for the methods described.