Solution of second order non-linear boundary value problems by the boundary element method
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In this work a procedure has been developed which makes it possible to use the boundary element method to solve non-linear partial differential equations of the second order in two variables. The problems solved are boundary value problems, and the non-linearity can be of any kind. A mathematical formulation is given which makes it unnecessary to have a knowledge of the fundamental solution for every type of non-linear equation. For second order partial differential equations, the use of Green's function for the Laplace equation as the fundamental solution has proved adequate to solve a wide variety of non-linear problems. Newton Raphson method was used to solve the algebraic non-linear equations resulting from a discretization of the boundary integral equation;A method is also presented to improve the grid once the solution has been found on an existing grid. This method has proved to be very efficient in the solution of non-linear problems. The method uses some of the features unique to the boundary element method to calculate the local errors. The problem is solved on the refined grid, and, if necessary, the grid is refined repeatedly until some convergence criterion is satisfied;A grid generation algorithm which borrows heavily from the grid optimization process has been developed. Using an initial coarse grid, the elements are broken up into smaller elements until a grid of desired fineness is obtained. The algorithm developed can handle regions enclosed by curved boundaries, the only stipulation being that the curved boundaries have a parametric representation. The grid generation algorithm also generates boundary elements by using a procedure similar to the boundary optimization scheme developed earlier.