Study of finite element analysis on differential manifolds

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Suppose that we are interested in an elliptic boundary value problem defined on a bounded domain. In this research, we consider effects of perturbation of the domain to the solution of the boundary value problem. In particular, we have established a scheme of computing the first and second variations of functionals defined with the solution of BVP. Using the scheme, we have found an alternative proof of the classical Hadamard's variational formula. We also apply the scheme to the research of a free boundary problem called the "dam problem". We have obtained the first and second variation of the functional which governs the dam problem. Using the obtained the first variation, we propose a new iterative algorithm for numerical solution of the dam problem.