Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

Mesh Adaptivity and Artificial di usion in Hyperbolic Problems

Charalambos Makridakis

Department of Applied Mathematics at University of Crete

Abstract:   We consider finite element, finite di erence schemes and adaptive strategies for the approximation of nonlinear hyperbolic systems of conservation laws. It is known that finite elements are not a very popular choice for computing singular solutions of hyperbolic problems. When applied directly to the system they will result computational solutions with oscillatory character close to shocks and/or not convergent approximations. This well known phenomenon is of course related to the fact that direct finite element discretizations behave like dispersion approximations. Similar behavior is observed in the study of related dispersive di erence schemes approximating conservation laws. To overcome this diffculty in using standard schemes, several modifications have been proposed in the literature by adding artificial viscosity and/or extra stabilization terms in the schemes. The higher order versions of these methods are complicated and with poor theoretical backup. Recently many of these schemes have been tested with various mesh adaptation methods. Our motivation was to consider schemes designed to be used in conjunction with appropriate mesh refinement. We will show that mesh refinement strategies can change our view on the application of many of the known schemes. This is because the mesh distribution influences not only the accuracy of the scheme but also its stability behavior. In this talk we consider new finite element schemes for HCL designed to be used with mesh adaptivity; discuss adaptive strategies for shock computations based on estimator functionals or a posteriori error control; conclude to new and rather unexpected observations for the behavior of Entropy Conservative di erence schemes. Classical dispersive-type finite element and finite difference schemes are also considered. We then try to explain why many of these schemes when used in conjunction with appropriate adaptive strategies yield computational solutions with surprisingly stable behavior. We conclude that the qualitative e ect of mesh adaptivity is the responsible mechanism and not the 'resolution' of boundary layers/shocks. These conclusions are also applicable to convection - di usion problems where similar numerical problems occur.