- New adaptive finite element methods are presented for the analysis of unsteady inviscid compressible flow in arbitrary two-dimensional domains. The procedures described herein are used in conjunction with a semi-explicit two-step algorithm for solving the time-dependent Euler equations in two space dimensions. Two schemes are presented for monitoring the evolution of error, and error estimates are used as a basis for a mesh refinement strategy. The capability of unrefinement (adaptively coarsening the mesh) is also included. The methods do not require a structured mesh and are applicable to quite general geometries. 1986.