Adaptive finite element methods for conservation laws

Abstract

The purpose of this lecture is to give an introductory overview of our recent work together with coworkers on computational methods for conservation laws, which are reliable in the sense that the computational error may be controled on a given tolerance level in a given norm, and efficient in the sence that the desired error control may be achieved at (nearly) minimal computational cost. To satisfy the desired criteria of reliability and efficiency, the computational methods are adaptive with feed back from the computational process. The adaptive methods are based on a posteriori error estimates, where the error is estimated in terms of the mesh size, the residual of the computed solution, and certain stability factors measuring relevant stability properties of the solution being approximated through the solution of an associated linearized dual problem. The a posteriori error estimates give stopping criteria guaranteeing the desired error control, and also serve as part of the modification criteria for adaptively choosing the computational mesh. We prove analytically that the stability factors in the basic model cases of shocks and rarefactions in one dimension, are of moderate size. We also present results from numerical computations of dual solutions and stability factors.