Numerical solution of the biharmonic equation
Petter E. Bjørstad · Defense Technical Information Center (DTIC) · 1980
The numerical solution of discrete approximations to the first biharmonic boundary value problem in rectangular domains is studied. Several finite difference schemes are compared and a family of new fast algorithms for the solution of the discrete systems is developed. These methods are optimal, having a theoretical computational complexity of 0(N2) arithmetic operations and requiring N2+0(N) storage locations when solving the problem on an N by N grid. Several practical computer implementations of the algorithm are discussed and compared. These implementations require a(sq N)+ b(sq N)logN airthmetic operations with ba. The algorithms take full advantage of vector or parallel computers and can also be used to solve a sequence of problems efficiently. A new fast direct method for the biharmonic problem on a disk is also developed. It is shown how the new method of solution is related to the associated eigenvalue problem. The results of extensive numerical tests and comparisons are included throughout the dissertation. It is believed that the material presented provides a good foundation for practical computer implementations and that the numerical solution of the biharmonic equation in rectangular domains from now on, will be considered no more difficult than Poisson's equation. (Author)