| uch attention has been devoted to biharmonic problems because of their occurrence in many applications. For example, in the aviation industry, where flow and elasticity problems are to be solved, the biharmonic problem is of fundamental importance. Various methods, usually finite difference or finite element Galerkin methods, have been developed for solving the biharmonic problem numerically. In this dissertation, we present an orthogonal spline collocation (OSC) method with piecewise Hermite bicubics to discretize the biharmonic Dirichlet problem: ;In order to employ the capacitance matrix method, we select the OSC approximation of the auxiliary problem: ;We also derive some theoretical results. In particular, we prove existence and uniqueness of the OSC solution of the biharmonic problem: ;We present results of some numerical experiments which verify the fourth order accuracy of the approximations, and, in particular, the superconvergence of the derivative approximations at the mesh points. We compare a sequential implementation of our algorithm with that of Bjorstad, which has an optimal cost of ;Finally, we extend our algorithm to the biharmonic problem:... |
