Adaptive least squares finite element method

In this paper a new error estimator for the compressible shocked now problems to be used in conjunction with the least-squares finite element method is presented. The error estimator is shown to be naturally related to the least-squares method, providing a suitable measure of the errors in the solution. The estimator is easily calculated by the use of already existing matrices of the least-squares computation and hence is very cheap. Furthermore, the estimator dose not need the evaluation of the second order derivatives which further reduces the computational effort. The estimator is used with conjunction with a remeshing procedure and shown to be very effective. The ability of the remeshing procedure to incorporate one dimensional features of the solution by introducing stretched elements and producing unstructured meshes with significant variation in mesh spacing makes it a suitable choice for adaptive refinement of the shocked now solutions. The application of the presented adaptive refinement technique to some shocked now problem is presented to show the flexibility and effectiveness of both the error estimator and the remeshing algorithm used.