A hybrid finite element-alternating direction implicit method for solving parabolic differential equations on multilayers with irregular geometry

A hybrid finite element-alternating direction implicit (ADI) method has been developed for solving three-dimensional parabolic differential equations with irregular geometry in the xy-directions. It is shown that this scheme is unconditionally stable. A domain decomposition algorithm that employs the hybrid finite element-ADI scheme is then developed to solve three-dimensional parabolic differential equations on multilayers with irregular geometry. In the algorithm, the generalized divide-and-conquer method for solving tridiagonal linear systems is applied in order to overcome the problem with the unknown value at the interface between layers. Finally, we tested the accuracy of the hybrid finite element-ADI method, and applied the domain decomposition algorithm to obtain temperature distributions and to ascertain rises in temperature for an X-ray irradiated photoresist with a gear pattern on the mask.