Error estimation for fourth order partial differential equations.

摘要

The goal of this dissertation is to find a posteriori error estimates for fourth order two-point boundary value problems based on Hermite-Lobatto interpolation error estimates.; In this dissertation, the motivation for investigating fourth order partial differential equations is outlined first. This is necessary because solving these partial differential equations has been much less examined compared to the extensive studies that have been done for the second order partial differential equations. A brief overview is also provided on what has been achieved in the numerical analysis of fourth order partial differential equations.; Next the finite element theory for fourth order partial differential equations is introduced. The discussions are limited to linear problems with homogeneous boundary conditions. The hierarchic approach is applied for building a set of Hermite-Lobatto basis functions.; Interpolation error estimates are obtained from an extension of the error formula for the Hermite-Lobatto interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and H 2 seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness.; The most important properties of the Hermite-Lobatto polynomials have been investigated. Some of them provide tools to assemble the finite element algorithm such as the recurrence relationship property. Others guarantee numerical stability such as the orthogonality property.; Future research to be done by the author is discussed.