Simplification and error analysis for moving finite element methods.

摘要

In this thesis, the one dimensional moving finite element (MFE) scheme of Miller is analyzed and simplified.;We show how the MFE scheme can lead to a decoupled system of nonlinear ordinary differential equations for node placement and corresponding amplitude of approximate solution.;For a scheme with penalty terms, the simplified MFE scheme leads to nonlinear ordinary differential system with respect to mesh points and a separate system of differential equations related to solution values at each mesh point.;We also establish simplified scheme for Gradient Weighted Moving Finite Element method. The resulting ordinary differential equations are completely decouple, and partly decouple when penalty terms are added into the scheme.;The error, analysis for application of MFE scheme to linear partial differential equations is discussed. An a posteriori error estimate is derived. It provides insight into overall accuracy of the approximate solution.;We also combine MFE with the moving mesh method of Russell. Specifically, we couple the equation for mesh points from Russell's method with the one for solution of PDE in simplified MFE. This combination allows for the application of the MFE scheme without an explicit selection of a penalty function.;Finally, results from a set of numerical experiments are presented. These demonstrate both the reduced computational cost and improved stability of the simplified MFE method.