Fundamentals of Numerical Analysis

In Chap. 5, covering several boundary value problems of elliptic partial differential equations, we saw that the existence of their unique solutions could be guaranteed using solutions of the weak form. These will be referred to as the exact solutions. Exact solutions can be found analytically if the shape of the domain is somewhat simple such as a rectangle or an ellipse. However, difficulties arise for domains whose shape may have arbitrarily moved as examined in this book. In order to solve a shape optimization problem, even if an exact solution is not possible, one can resort to a numerical analysis method to obtain an approximate solution.