Minimal surfaces existence, uniqueness and computation

Abstract

The theoretical background of the classical minimal surface equation has been laid out in the first part of the thesis. In the second part, the minimal surface equation has been solved numerically by using the Finite Difference Method and Finite Element Method. The algorithmic procedure for the Finite Difference Method is to find the solution of each point by using the initial guess as zeros inside the curve and boundary values on the boundary points. The idea of the Finite Element Method is to first look for the variational formulation of the problem and to formulate system on each element, then get the assembly of systems of each element into a system on the whole domain and use the iterative method or direct method to get the solution. By using the idea of D.Gilbarg and N.S.Trudinger, I show the solutions to the minimal surface equation with the given boundary conditions are bounded. This is necessary for numerical method to work. Moreover, the drawbacks of both methods are discussed at the end of both methods. The numerical solution by Finite Difference Method has been compared with the numerical one by Finite Element Method. Algorithmic programs and their outputs have been appeared as appendix.