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Finite Element Solutions to Nonlinear Partial Differential Equations (open access)

Finite Element Solutions to Nonlinear Partial Differential Equations

This paper develops a numerical algorithm that produces finite element solutions for a broad class of partial differential equations. The method is based on steepest descent methods in the Sobolev space H¹(Ω). Although the method may be applied in more general settings, we consider only differential equations that may be written as a first order quasi-linear system. The method is developed in a Hilbert space setting where strong convergence is established for part of the iteration. We also prove convergence for an inner iteration in the finite element setting. The method is demonstrated on Burger's equation and the Navier-Stokes equations as applied to the square cavity flow problem. Numerical evidence suggests that the accuracy of the method is second order,. A documented listing of the FORTRAN code for the Navier-Stokes equations is included.
Date: August 1981
Creator: Beasley, Craig J. (Craig Jackson)
System: The UNT Digital Library
Geometric Problems in Measure Theory and Parametrizations (open access)

Geometric Problems in Measure Theory and Parametrizations

This dissertation explores geometric measure theory; the first part explores a question posed by Paul Erdös -- Is there a number c > 0 such that if E is a Lebesgue measurable subset of the plane with λ²(E) (planar measure)> c, then E contains the vertices of a triangle with area equal to one? -- other related geometric questions that arise from the topic. In the second part, "we parametrize the theorems from general topology characterizing the continuous images and the homeomorphic images of the Cantor set, C" (abstract, para. 5).
Date: August 1981
Creator: Ingram, John M. (John Michael)
System: The UNT Digital Library
The Steepest Descent Method Using Finite Elements for Systems of Nonlinear Partial Differential Equations (open access)

The Steepest Descent Method Using Finite Elements for Systems of Nonlinear Partial Differential Equations

The purpose of this paper is to develop a general method for using Finite Elements in the Steepest Descent Method. The main application is to a partial differential equation for a Transonic Flow Problem. It is also applied to Burger's equation, Laplace's equation and the minimal surface equation. The entire method is tested by computer runs which give satisfactory results. The validity of certain of the procedures used are proved theoretically. The way that the writer handles finite elements is quite different from traditional finite element methods. The variational principle is not needed. The theory is based upon the calculation of a matrix representation of operators in the gradient of a certain functional. Systematic use is made of local interpolation functions.
Date: August 1981
Creator: Liaw, Mou-yung Morris
System: The UNT Digital Library