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Title:  Unified Multilevel Adaptive Finite Element Methods for Elliptic Problems 
Author(s):  Mitchell, William F. 
Doctoral Committee Chair(s):  Skeel, Robert D. 
Department / Program:  Computer Science 
Discipline:  Computer Science 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics
Computer Science 
Abstract:  Many elliptic partial differential equations can be solved numerically with near optimal efficiency through the uses of adaptive refinement and multigrid solution techniques. It is our goal to develop a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements. The basic step of the refinement process is the bisection of a pair of triangles, which corresponds to the addition of one or more basis functions to the approximation space. An approximation of the resulting change in the solution is used as an error indicator to determine which triangles to divide. The multigrid iteration uses a redblack GaussSeidel relaxation in which the black relaxations are used only locally. The grid transfers use the change between the nodal and hierarchical bases. This multigrid iteration requires only O(N) operations, even for highly nonuniform grids, and is defined for any finite element space. The full multigrid method is an optimal blending of the processes of adaptive refinement and multigrid iteration. So as to minimize the number of operations required, the duration of the refinement phase is based on increasing the dimension of the approximation space by some fixed factor which is determined to be the largest possible for the given errorreducing power of the multigrid iteration. The result is an algorithm which (i) uses only O(N) operations with a reasonable constant of proportionality, (ii) solves the discrete system to the accuracy of the discretization error, (iii) is able to achieve the optimal order of convergence of the discretization error in the presence of singularities. Numerical experiments confirm this for linear, quadratic and cubic elements. It is believed that the method can also be applied to more practical problems involving systems of PDE's, time dependence, and three spatial dimensions. 
Issue Date:  1988 
Type:  Text 
Description:  119 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1988. 
URI:  http://hdl.handle.net/2142/69604 
Other Identifier(s):  (UMI)AAI8908782 
Date Available in IDEALS:  20141215 
Date Deposited:  1988 
This item appears in the following Collection(s)

Dissertations and Theses  Computer Science
Dissertations and Theses from the Dept. of Computer Science 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois