Bi-quartic spline collocation methods for fourth-order boundary value problems with an application to the biharmonic Dirichlet problem.
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文摘
We present optimal bi-quartic spline collocation methods for the numerical solution of fourth-order boundary value problems on rectangular domains, and apply a particular instance of these methods to the biharmonic Dirichlet problem.;The methods are based on quartic splines and the collocation discretization methodology with midpoints of a uniform partition as the collocation points. While the standard quartic spline method usually provides second-order approximations, the two bi-quartic spline collocation methods developed in this thesis are observed to produce approximations which are of sixth order at grid points and midpoints, and of fifth order at other points. Both are based on high order perturbations of the differential and boundary conditions operators. The one-step (extrapolated) method forces the approximations to satisfy a perturbed problem, while the three-step (deferred-correction) method proceeds in three steps and perturbs the right sides of the linear equations appropriately.;The three-step bi-quartic spline collocation method is then applied to the biharmonic Dirichlet problem and a fast solver for the resulting linear systems is developed. The fast solver consists of solving an auxiliary biharmonic problem with Dirichlet and second derivative boundary conditions along the two opposite boundaries, and a one-dimensional problem related to the two opposite boundaries. The linear system arising from the bi-quartic spline collocation discretization of the auxiliary problem is solvable by fast Fourier transforms. The problem related to the two opposite boundaries is solved by preconditioned GMRES (PGMRES). We present a detailed analysis of the performance of PGMRES by studying bounds for the eigenvalues of the preconditioned matrix. We show that the number of PGMRES iterations is independent of the grid size n. As a result, the complexity of the fast solver is O(n 2log(n)). Numerical experiments from a variety of problems, including practical applications and problems more general than the analysis assumes, verify the accuracy of the discretization scheme and the effectiveness of the fast solver.