Adaptive Finite Element Methods for Parabolic Interface Problems

Abstract

"The main objective of this thesis is to study adaptive nite element methods (AFEMs) for parabolic interface problems in a bounded convex polygonal domain in R2. Interface problems arise in a wide variety of applications in science and engineering such as material sciences and uid dynamics when two or more distinct materials or uids with di erent conductivities or densities or di usions are interacting across the interface. Due to the discontinuity of the coe cients along the interface, the analytic solutions are rarely available for the interface problems. Therefore, the numerical approximation is the only way to proceed with such problems. Even if the solution is smooth in each individual subdomain, the global regularity of the solution of such problem is very low. As a result, it is very challenging to achieve higher order accuracy in the nite element method (FEM). Therefore, much attention has been paid in the recent years to the study both theory and numerics of time-dependent interface problems. It is known that AFEMs are widely used numerical techniques to enhance the accuracy and e ciency of the nite element method. The key to the success of AFEMs relies on the a posteriori error analysis, which provides error indicators for the design of adaptive algorithms. The adaptive method reduces the computational e orts and ensures higher density nodes in a particular area of the given domain where the solution is very di cult to approximate."