Abstract
We develop, analyze, and test a new discontinuous finite element method (DFEM), namely the piece-wise linear discontinuous (PWLD) method for transport problems. We compare the PWLD method against the rational finite element (RFE) method. We show that the RFE method on rectangles reduces to the well-known bilinear discontinuous (BLD) method, which performs well on rectangles. We show that the coefficients of the PWLD method can easily be evaluated analytically for general polygons, whereas, the RFE method requires numerical integration for general polygons. We perform an asymptotic analysis to study the behavior of the PWLD and the RFE methods in the optically thick and diffusive regions. We show that the leading-order RFE solution on rectangles satisfies an analogous continuous RFE discretization of the diffusion equation. We also show that the leading-order PWLD solution satisfies a very reasonable discretization of the diffusion equation. We compare the linear discontinuous (LD), BLD, fully lumped BLD, PWLD, fully lumped PWLD methods on a series of test problems in XY geometry. The numerical results show that the PWLD method performs as well as the BLD method on rectangles. The results also show that the lumping of the PWLD equations adds robustness of the method, much like it does with the BLD method. We conclude that the PWLD method has potential to produce reasonable solution in thick diffusive region with arbitrary polygonal cells. Numerical testing of the PWLD and the RFE methods with arbitrary polygons remains as a potential for further research.
Stone, Hiromi (2002). Discontinuous finite element methods for particle transport problems. Master's thesis, Texas A&M University. Available electronically from
https : / /hdl .handle .net /1969 .1 /ETD -TAMU -2002 -THESIS -S754.