| Summary: | In this thesis, we discuss the development, implementation and testing of a piecewise
linear (PWL) continuous Galerkin finite element method applied to the threedimensional
diffusion equation. This discretization is particularly interesting because it
discretizes the diffusion equation on an arbitrary polyhedral mesh. We implemented our
method in the KULL software package being developed at Lawrence Livermore
National Laboratory. This code previously utilized Palmer's method as its diffusion
solver, which is a finite volume method that can produce an asymmetric coefficient
matrix. We show that the PWL method produces a symmetric positive definite
coefficient matrix that can be solved more efficiently, while retaining the accuracy and
robustness of Palmer's method. Furthermore, we show that in most cases Palmer's
method is actually a non-Galerkin PWL finite element method.
Because the PWL method is a Galerkin finite element method, it has a firm theoretical
background to draw from. We have shown that the PWL method is a well-posed
discrete problem with a second-order convergence rate. We have also performed a
simple mode analysis on the PWL method and Palmer's method to compare the accuracy
of each method for a certain class of problems.
Finally, we have run a series of numerical tests to uncover more properties of both the
PWL method and Palmer's method. These numerical results indicate that the PWL
method, partially due to its symmetric matrix, is able to solve large-scale diffusion
problems very efficiently.
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